Extended catching scale

Extended catching scale (exCS) is an extension of catching scale. In catching scale, {c 1`2} fails because {c #} usually reduces to separators with higher level than the grave accent. In exCS, what {c #} reduces to has limited level, and {c #} itself also has limited level, less than the grave accent. exCS puts separators with c into the process, gives them levels, so that higher separators, such as the grave accent and multiple comma can act on them.

Levels

Generally, if lv({#}) < lv(`) and lv({#′}) < lv(`), then lv({#}) < lv({c #′}) < lv(`), where # and #′ don’t begin with “c”. Comparisons between {c #} can be defined as:

  • If lv({#1}) > lv({#2}) then lv({c #1}) > lv({c #2}); if lv({#1}) < lv({#2}) then lv({c #1}) < lv({c #2}); if lv({#1}) = lv({#2}) then lv({c #1}) = lv({c #2}).

Note that {c #} is valid only if lv({#}) < lv(`).

We say a separator A is a c-free if the symbol “c” doesn’t appear anywhere in the expression of A.

Case of c in the process

Let S = {A is separator|A is c-free∧ λx.s(x,x A 2) ≈ λx.ss(x,x A 2)} be the set of catching separators.

Adding this case to the main process forms exCS.

  • Case: If the entry n is not 1, and the {c 1} is immediately before you, then
    1. Let separator M1 = max{A∈S|¬∃B∈S(λx.s(x,x A 2) >* λx.s(x,x B 2))} where the max is by level comparison, and Mi+1 = {1 Mi 2}.
    2. Find minimum of t such that lv(Mt) < lv(`).
    3. Change the “{c 1} n” into “Mt 2 {c 1} n-1″.
    4. The process ends.
  • Case: If the entry n is not 1, and the “{c” is immediately before you, then
    1. Let separator M1 = max{A∈S|λx.s(x,x A 2) >* λx.s(x,x {c n-1 #} 2)∧ ¬∃B∈S(λx.s(x,x A 2) >* λx.s(x,x B 2) >* λx.s(x,x {c n-1 #} 2))} where the max is by level comparison, and Mi+1 = {1 Mi 2}.
    2. Find minimum of t such that lv(Mt) < lv(`).
    3. Change the “{c n #}” into “Mt“.
    4. The process ends.

Explanation

The M1 is identical to the separators in catching scale, but with a high level. Step 2 and 3 of the case of “c” makes a separator in the form {1{1…{1 M1 2}…2}2} to get lower level than the grave accent.

Now {c 1`2} reduces to {c 1…{c 1 {c 1,2} 2}…2}, which reduces to {c 1…{c 1 {1{1,,1,2}2} 2}…2}. Also, {c #} has higher level than what it reduces to. So in exCS the loop problem in catching scale is solved.

Comparison

Here’re some “separator reduction” in exCS.

{c 1,2} = {1{1,,1,,2}2}
{c 1{c 1,2}2} = {1{1,,1,,3}2}
{c 1{c 1{c 1,2}2}2} = {1{1,,1,,4}2}
{c 1`2} = {1{1,,1,,1,2}2}
{c 2`2} = {1{1,,1,,1{1,,1,2}2}2}
{c 1,2`2} = {1{1,,1,,1{1,,1{1,,1,,2}2}2}2}
{c 1{c 1,2}2`2} = {1{1,,1,,1{1,,1{1,,1,,3}2}2}2}
{c 1{c 1{c 1,2}2}2`2} = {1{1,,1,,1{1,,1{1,,1,,4}2}2}2}
{c 1{c 1`2}2`2} = {1{1,,1,,1{1,,1{1,,1,,1,2}2}2}2}
{c 1{c 1`2}3`2} = {1{1,,1,,1{1,,1{1,,1,,1{1,,1{1,,1,,1,2}2}2}2}2}2}
{c 1{c 1`2}1,2`2} = {1{1,,1,,1{1,,1,,2}2}2}
{c 2{c 1`2}1,2`2} = {1{1,,1{1,,1,,1{1,,1,,2}2}1{1,,1,2}2}2}
{c 1{c 1`2}1,1,2`2} = {1{1,,1{1,,1,,1{1,,1,,2}2}1{1,,1,,2}2}2}
{c 1{2{1,,1,,1,2}2}2`2} = {1{1,,1{2,,1,,1{1,,1,,2}2}2}2}
{c 1{1`2{1,,1,,1,2}2}2`2} = {1{1{1,,2,,2}2{1,,1,,1{1,,1,,2}2}2,,1,,2}2}
{c 1{1{1,,1,,1,2}1,2}2`2} = {1{1{1,,1,,1{1,,1,,2}2}1,2,,1,,2}2}
{c 1{1{1{1,,1,2}2,,1,,1,2}2}2`2} = {1{1{1,,1,2,,2}2,,1,,1{1,,1,,2}2}2}
{c 1{1{1{1,,1,,2}2,,1,,1,2}2}2`2} = {1{1{1,,1,,3}2,,1,,1{1,,1,,2}2}2}
{c 1{1{1{1,,1,,1,2}2,,1,,1,2}2}2`2} = {1{1{1,,1,,1,2}2,,1,,1{1,,1,,2}2}2}
{c 1{1{1{1,,1,,1,2}2,,1,,1,2}2}1,2`2} = {1{1{1,,1,,1{1,,1,,2}2}2,,1,,1{1,,1,,2}2}2}
{c 1{1{1,,2,,1,2}2}2`2} = {1{1,,2,,1{1,,1,,2}2}2}
{c 1{1{1,,1,2,,1,2}2}2`2} = {1{1,,1,2,,1{1,,1,,2}2}2}
{c 1{1{1,,1,,2,2}2}2`2} = {1{1,,1,,2{1,,1,,2}2}2}
{c 1{1{1,,1,,1,3}2}2`2} = {1{1,,1,,1,2{1,,1,,2}2}2}
{c 1{1{1,,1,,1,3}2}1,2`2} = {1{1,,1,,1{1,,1,,2}3}2}
{c 1{1{1,,1,,1{2}2}2}2`2} = {1{1,,1,,1{2,,1,,2}2}2}
{c 1{1{1,,1,,1{1`2}2}2}2`2} = {1{1,,1,,1{1{1,,2,,2}2,,1,,2}2}2}
{c 1{1{1,,1,,1{1{1,,1,,1,2}2}2}2}2`2} = {1{1,,1,,1{1{1,,1,,1,2}2,,1,,2}2}2}
{c 1{1{1,,1,,1`2}2}2`2} = {1{1,,1,,1{1,,2,,2}2}2}
{c 1{1{1,,1,,1{1,,1,2}2}2}2`2} = {1{1,,1,,1{1,,1,2,,2}2}2}
{c 1{c 1,2`2}2`2} = {1{1,,1,,1{1,,1{1,,1,,3}2,,2}2}2}
{c 1{1{1,,1,,1{1,,1,,2}2}2}2`2} = {1{1,,1,,1{1,,1,,3}2}2}
{c 1{c 1{c 1,2`2}2`2}2`2} = {1{1,,1,,1{1,,1{1,,1,,4}2,,3}2}2}
{c 1{c 1{c 1{c 1,2`2}2`2}2`2}2`2} = {1{1,,1,,1{1,,1{1,,1,,5}2,,4}2}2}
{c 1`3} = {1{1,,1,,1{1,,1,,1,2}2}2}
{c 1,2`3} = {1{1,,1,,1{1,,1,,1{1,,1{1,,1,,2}2}2}2}2}
{c 1{c 1`3}1,2`3} = {1{1,,1,,1{1,,1,,1{1,,1,,2}2}2}2}
{c 1{c 1,2`3}2`3} = {1{1,,1,,1{1,,1,,1{1,,1{1,,1,,3}2,,2}2}2}2}
{c 1{c 1{c 1,2`3}2`3}2`3} = {1{1,,1,,1{1,,1,,1{1,,1{1,,1,,4}2,,3}2}2}2}
{c 1`4} = {1{1,,1,,1{1,,1,,1{1,,1,,1,2}2}2}2}
{c 1`5} = {1{1,,1,,1{1,,1,,1{1,,1,,1{1,,1,,1,2}2}2}2}2}
{c 1`1,2} = {1{1,,1,,1,,2}2}
{c 2`1,2} = {1{1,,1,,1{1,,1,,1,,2}1{1,,1,2}2}2}
{c 1{c 1`1,2}1,2`1,2} = {1{1,,1,,1{1,,1,,1,,2}1{1,,1,,2}2}2}
{c 1{1`2{1,,1,,1,,2}2}2`1,2} = {1{1{1,,2,,2}2{1,,1,,1{1,,1,,1,,2}1{1,,1,,2}2}2,,1,,2}2}
{c 1{1{1,,1,,1,,2}1`2}2`1,2} = {1{1{1,,1,,1{1,,1,,1,,2}1{1,,1,,2}2}1{1,,2,,2}2,,1,,2}2}
{c 1{1{1{1,,1,2}2,,1,,1{1,,1,,1,,2}2}2}2`1,2} = {1{1{1,,1,2,,2}2,,1,,1{1,,1,,1,,2}1{1,,1,,2}2}2}
{c 1{1{1{1,,1,,1,2}2,,1,,1{1,,1,,1,,2}2}2}2`1,2} = {1{1{1,,1,,1,2}2,,1,,1{1,,1,,1,,2}1{1,,1,,2}2}2}
{c 1{1{1,,2,,1{1,,1,,1,,2}2}2}2`1,2} = {1{1,,2,,1{1,,1,,1,,2}1{1,,1,,2}2}2}
{c 1{1{1,,1,,2{1,,1,,1,,2}2}2}2`1,2} = {1{1,,1,,2{1,,1,,1,,2}1{1,,1,,2}2}2}
{c 1{1{1,,1,,1{1,,1,,1,,2}3}2}2`1,2} = {1{1,,1,,1{1,,1,,1,,2}2{1,,1,,2}2}2}
{c 1{1{1,,1,,1{1,,1,,1,,2}1,2}2}2`1,2} = {1{1,,1,,1{1,,1,,1,,2}1{1,,1,,2}1,2}2}
{c 1{1{1,,1,,1{1,,1,,1,,2}1`2}2}2`1,2} = {1{1,,1,,1{1,,1,,1,,2}1{1,,2,,2}2}2}
{c 1{c 1{c 1`1,2}1,2`1,2}2`1,2} = {1{1,,1,,1{1,,1,,1,,2}1{1,,1,,3}2}2}
{c 1{c 1{c 1{c 1`1,2}1,2`1,2}2`1,2}2`1,2} = {1{1,,1,,1{1,,1,,1,,2}1{1,,1,,4}2}2}
{c 1`2,2} = {1{1,,1,,1{1,,1,,1,,2}1{1,,1,,1,2}2}2}
{c 1`3,2} = {1{1,,1,,1{1,,1,,1,,2}1{1,,1,,1{1,,1,,1,2}2}2}2}
{c 1`1,3} = {1{1,,1,,1{1,,1,,1,,2}1{1,,1,,1{1,,1,,1,,2}2}2}2}
{c 1`1,4} = {1{1,,1,,1{1,,1,,1,,2}1{1,,1,,1{1,,1,,1,,2}1{1,,1,,1{1,,1,,1,,2}2}2}2}2}
{c 1`1,1,2} = {1{1,,1,,1{1,,1,,1,,2}1{1,,1,,1,,2}2}2}
{c 2`1,1,2} = {1{1,,1,,1{1,,1,,1,,2}1{1,,1,,1,,2}1{1,,1,2}2}2}
{c 1`2,1,2} = {1{1,,1,,1{1,,1,,1,,2}1{1,,1,,1,,2}1{1,,1,,1,2}2}2}
{c 1`1,2,2} = {1{1,,1,,1{1,,1,,1,,2}1{1,,1,,1,,2}1{1,,1,,1{1,,1,,1,,2}2}2}2}
{c 1`1,1,3} = {1{1,,1,,1{1,,1,,1,,2}1{1,,1,,1,,2}1{1,,1,,1{1,,1,,1,,2}1{1,,1,,1,,2}2}2}2}
{c 1`1,1,1,2} = {1{1,,1,,1{1,,1,,1,,2}1{1,,1,,1,,2}1{1,,1,,1,,2}2}2}
{c 1`1{2}2} = {1{1,,1,,1{2,,1,,1,,2}2}2}
{c 1{c 1`1{2}2}1,2`1{2}2} = {1{1,,1,,1{1{1,,1,,2}2,,1,,1,,2}2}2}
{c 1{c 1{c 1`1{2}2}1,2`1{2}2}2`1{2}2} = {1{1,,1,,1{1{1,,1,,3}2,,1,,1,,2}2}2}
{c 1`2{2}2} = {1{1,,1,,1{1{1,,1,,1,2}2,,1,,1,,2}2}2}
{c 1`1{2}3} = {1{1,,1,,1{1{1,,1,,1{2,,1,,1,,2}2}2,,1,,1,,2}2}2}
{c 1`1{2}1,2} = {1{1{1,,1,,1,,2}2,,1,,1,,2}2}
{c 1`1{3}1,2} = {1{1{1,,1,,1,,2}3,,1,,1,,2}2}
{c 1`1{1,2}2} = {1{1{1,,1,,1,,2}1,2,,1,,1,,2}2}
{c 1`1{1{2}2}2} = {1{1{2,,1,,1,,2}2,,1,,1,,2}2}
{c 1`1{1{1,2}2}2} = {1{1{1{1,,1,,1,,2}1,2,,1,,1,,2}2,,1,,1,,2}2}
{c 1`1{1`2}2} = {1{1,,2,,1,,2}2}
{c 2`1{1`2}2} = {1{1{1,,2,,1,,2}1{1,,1,2}2,,1,,1,,2}2}
{c 1`2{1`2}2} = {1{1{1,,2,,1,,2}1{1,,1,,1,2}2,,1,,1,,2}2}
{c 1`1{1`2}3} = {1{1{1,,2,,1,,2}1{1,,1,,1{1,,2,,1,,2}2}2,,1,,1,,2}2}
{c 1`1{1`2}1,2} = {1{1{1,,2,,1,,2}1{1,,1,,1,,2}2,,1,,1,,2}2}
{c 1`1{1`3}2} = {1{1{1,,2,,1,,2}2{1,,1,,1,,2}2,,1,,1,,2}2}
{c 1`1{1`1,2}2} = {1{1{1,,2,,1,,2}1{1,,1,,1,,2}1,2,,1,,1,,2}2}
{c 1`1{1`1{1`2}2}2} = {1{1{1,,2,,1,,2}1{1{1,,2,,1,,2}2,,1,,1,,2}2,,1,,1,,2}2}
{c 1`1{1`1`2}2} = {1{1{1,,2,,1,,2}1{1,,2,,1,,2}2,,1,,1,,2}2}
{c 1`1{1`1`1`2}2} = {1{1{1,,2,,1,,2}1{1,,2,,1,,2}1{1,,2,,1,,2}2,,1,,1,,2}2}
{c 1`1{1{2`}2}2} = {1{1{2,,2,,1,,2}2,,1,,1,,2}2}
{c 1`1{1{1`2`}2}2} = {1{1{1,,2,,1,,2}2,,2,,1,,2}2}
{c 1`1{1{1{1`2`}2`}2}2} = {1{1{1{1,,2,,1,,2}2,,2,,1,,2}2,,2,,1,,2}2}
{c 1`1{1` `2}2} = {1{1,,3,,1,,2}2}
{c 1`1{1{1,,1,2}2}2} = {1{1,,1,2,,1,,2}2}
{c 2`1{1{1,,1,2}2}2} = {1{1,,1{1,,1,2}2,,1,,2}2}
{c 1`2{1{1,,1,2}2}2} = {1{1,,1{1,,1,,1,2}2,,1,,2}2}
{c 1`1{1{1,,1,2}2}1,2} = {1{1,,1{1,,1,,1,,2}2,,1,,2}2}
{c 1`1{1{1,,2,2}2}2} = {1{1,,2{1,,1,,1,,2}2,,1,,2}2}
{c 1`1{1{1,,1{2}2}2}2} = {1{1,,1{2,,1,,1,,2}2,,1,,2}2}
{c 1`1{1{1,,1{1`2}2}2}2} = {1{1,,1{1{1,,2,,1,,2}2,,1,,1,,2}2,,1,,2}2}
{c 1`1{1{1,,1`2}2}2} = {1{1,,1{1,,2,,1,,2}2,,1,,2}2}
{c 1`1{1{1,,1{1,,1,2}2}2}2} = {1{1,,1{1,,1,2,,1,,2}2,,1,,2}2}
{c 1`1{1{1,,1,,2}2}2} = {1{1,,1,,2,,2}2}
{c 1`1{1{1,,1,,3}2}2} = {1{1,,1,,3,,2}2}
{c 1`1{1{1,,1,,1,2}2}2} = {1{1,,1,,1,2,,2}2}
{c 1`1{c 1`1,2}2} = {1{1,,1,,1,,3}2}
{c 1`1{c 1`1{c 1`1,2}2}2} = {1{1,,1,,1,,4}2}
{c 1`1`2} = {1{1,,1,,1,,1,2}2}
{c 2`1`2} = {1{1,,1,,1,,1{1,,1,2}2}2}
{c 1{c 1`1`2}1,2`1`2} = {1{1,,1,,1,,1{1,,1,,2}2}2}
{c 1`2`2} = {1{1,,1,,1,,1{1,,1,,1,2}2}2}
{c 1`1{c 1`1`2}1,2`2} = {1{1,,1,,1,,1{1,,1,,1,,2}2}2}
{c 1`1`3} = {1{1,,1,,1,,1{1,,1,,1,,1,2}2}2}
{c 1`1`1,2} = {1{1,,1,,1,,1,,2}2}
{c 1`1`2,2} = {1{1,,1,,1,,1{1,,1,,1,,1,,2}1{1,,1,,1,,1,2}2}2}
{c 1`1`1,1,2} = {1{1,,1,,1,,1{1,,1,,1,,1,,2}1{1,,1,,1,,1,,2}2}2}
{c 1`1`1{2}2} = {1{1,,1,,1,,1{2,,1,,1,,1,,2}2}2}
{c 1`1`1{1`2}2} = {1{1,,2,,1,,1,,2}2}
{c 1`1`1{1{1,,1,2}2}2} = {1{1,,1,2,,1,,1,,2}2}
{c 1`1`1{c 1`1`1,2}2} = {1{1,,1,,1,,1,,3}2}
{c 1`1`1`2} = {1{1,,1,,1,,1,,1,2}2}
{c 1`1`1`1`2} = {1{1,,1,,1,,1,,1,,1,2}2}
{c 1{2`}2} = {1{1{2,,}2}2}
{c 2{2`}2} = {1{1{1{1,,1,2}2,,}2}2}
{c 1{c 1{2`}2}1,2{2`}2} = {1{1{1{1,,1,,2}2,,}2}2}
{c 1`2{2`}2} = {1{1{1{1,,1,,1,2}2,,}2}2}
{c 1`1`2{2`}2} = {1{1{1{1,,1,,1,,1,2}2,,}2}2}
{c 1{2`}3} = {1{1{1{1{2,,}2}2,,}2}2}
{c 1{2`}1,2} = {1{1{1,,2,,}2}2}

Let separator ◊ = {1{1{1,,2,,}2}2,,}, then

{c 2{2`}1,2} = {1{1◊1{1,,1,2}2}2}
{c 1{c 1{2`}1,2}1,2{2`}1,2} = {1{1◊1{1,,1,,2}2}2}
{c 1`2{2`}1,2} = {1{1◊1{1,,1,,1,2}2}2}
{c 1`1`2{2`}1,2} = {1{1◊1{1,,1,,1,,1,2}2}2}
{c 1{2`}2,2} = {1{1◊1{1{2,,}2}2}2}
{c 1{2`}1,3} = {1{1◊1{1◊2}2}2}
{c 1{2`}1,1,2} = {1{1◊1,,2}2}
{c 1{2`}2,1,2} = {1{1◊1{1◊1,,2}1{1{2,,}2}2}2}
{c 1{2`}1,1,1,2} = {1{1◊1{1◊1,,2}1{1◊1,,2}2}2}
{c 1{2`}1{2}2} = {1{1◊1{2◊1,,2}2}2}
{c 1{2`}1{1`2}2} = {1{1,,2◊1,,2}2}
{c 1{2`}1{1{1{2,,}2}2}2} = {1{1{2,,}2◊1,,2}2}
{c 2{2`}1{1{1{2,,}2}2}2} = {1{1{1{1,,1,2}2,,}2◊1,,2}2}
{c 1`2{2`}1{1{1{2,,}2}2}2} = {1{1{1{1,,1,,1,2}2,,}2◊1,,2}2}
{c 1{2`}2{1{1{2,,}2}2}2} = {1{1{1{1{2,,}2}2,,}2◊1,,2}2}
{c 1{2`}1,2{1{1{2,,}2}2}2} = {1{1{1{1◊2}2,,}2◊1,,2}2}
{c 1{2`}1{1{1{2,,}2}2}3} = {1{1{1{1◊1{1{2,,}2◊1,,2}2}2,,}2◊1,,2}2}
{c 1{2`}1{1{1{2,,}2}2}1,2} = {1{1{1{1◊1,,2}2,,}2◊1,,2}2}
{c 1{2`}2{1{1{2,,}2}2}1,2} = {1{1◊1{1{1{1◊1,,2}2,,}2◊1,,2}1{1{2,,}2}2}2}
{c 1{2`}1,2{1{1{2,,}2}2}1,2} = {1{1◊1{1{1{1◊1,,2}2,,}2◊1,,2}1{1◊2}2}2}
{c 1{2`}1{1{1{2,,}2}2}1,1,2} = {1{1◊1{1{1{1◊1,,2}2,,}2◊1,,2}1{1◊1,,2}2}2}
{c 1{2`}1{1{1{2,,}2}2}1{1{1{2,,}2}2}2} = {1{1◊1{1{1{1◊1,,2}2,,}2◊1,,2}1{1{2,,}2◊1,,2}2}2}
{c 1{2`}1{1{1{2,,}2}2}1{1{1{2,,}2}2}1,2} = {1{1◊1{1{1{1◊1,,2}2,,}2◊1,,2}1{1{1{1◊1,,2}2,,}2◊1,,2}2}2}
{c 1{2`}1{2{1{2,,}2}2}2} = {1{1◊1{2{1{1◊1,,2}2,,}2◊1,,2}2}2}
{c 1{2`}1{2{1{2,,}2}2}1,2} = {1{1{1◊1,,2}2{1{1◊1,,2}2,,}2◊1,,2}2}
{c 1{2`}1{1`2{1{2,,}2}2}2} = {1{1{1,,2◊1,,2}2{1{1◊1,,2}2,,}2◊1,,2}2}
{c 1{2`}1{1{1{2,,}2}3}2} = {1{1{1{1{2,,}2◊1,,2}2,,2◊1,,2}2{1{1◊1,,2}2,,}2◊1,,2}2}
{c 1{2`}1{1{1{1,,1,2}2{2,,}2}2}2} = {1{1{1,,1,2◊1,,2}2{1{1◊1,,2}2,,}2◊1,,2}2}
{c 1{2`}1{1{1{1{2,,}2}2{2,,}2}2}2} = {1{1{1{2,,}2◊1,,2}2{1{1◊1,,2}2,,}2◊1,,2}2}
{c 1{2`}1{1{1,,2{2,,}2}2}2} = {1{1,,2{1{1◊1,,2}2,,}2◊1,,2}2}
{c 1{2`}1{1{1{2,,}3}2}2} = {1{1{2,,}2{1{1◊1,,2}2,,}2◊1,,2}2}
{c 1{2`}1{1{1{2,,}3}2}1,2} = {1{1{1{1◊1,,2}2,,}3◊1,,2}2}
{c 1{2`}1{1{1{2,,}1,2}2}2} = {1{1{1{1◊1,,2}2,,}1,2◊1,,2}2}
{c 1{2`}1{1{1{2,,}1,2}2}1,2} = {1{1{1{1◊1,,2}2,,}1{1◊1,,2}2◊1,,2}2}
{c 1{2`}1{1{1{2,,}1,,2}2}2} = {1{1{1{1◊1,,2}2,,}1,,2◊1,,2}2}
{c 1{2`}1{1{1{2,,}1{2,,}2}2}2} = {1{1{1{1◊1,,2}2,,}1{2,,}2◊1,,2}2}
{c 1{2`}1{1{1{2,,}1{2,,}2}2}1,2} = {1{1{1{1◊1,,2}2,,}1{1{1◊1,,2}2,,}2◊1,,2}2}
{c 1{2`}1{1{1{3,,}2}2}2} = {1{1{2{1◊1,,2}2,,}2◊1,,2}2}
{c 1{2`}1{1{1{3,,}2}2}1,2} = {1{1{1{1◊1,,2}3,,}2◊1,,2}2}
{c 1{2`}1{1{1{1,2,,}2}2}2} = {1{1{1{1◊1,,2}1,2,,}2◊1,,2}2}
{c 1{2`}1{1{1{1`2,,}2}2}2} = {1{1{1{1,,2◊1,,2}2,,}2◊1,,2}2}
{c 1{2`}1{1{1{1{1{2,,}2}2,,}2}2}2} = {1{1{1{1{2,,}2◊1,,2}2,,}2◊1,,2}2}
{c 1{2`}1{c 1{2`}1,2}2} = {1{1◊2,,2}2}
{c 2{2`}1{c 1{2`}1,2}2} = {1{1◊1{1,,1,2}2,,2}2}
{c 1{2`}2{c 1{2`}1,2}2} = {1{1◊1{1{2,,}2}2,,2}2}
{c 1{2`}1,2{c 1{2`}1,2}2} = {1{1◊1{1◊2}2,,2}2}
{c 1{2`}1{c 1{2`}1,2}1,2} = {1{1◊1{1◊1,,2}2,,2}2}
{c 1{2`}1{1{1,,2◊2}2}2} = {1{1,,2◊1{1◊1,,2}2,,2}2}
{c 1{2`}1{1{1{2,,}2◊2}2}2} = {1{1{2,,}2◊1{1◊1,,2}2,,2}2}
{c 1{2`}1{1{1◊3}2}2} = {1{1◊2{1◊1,,2}2,,2}2}
{c 1{2`}1{1{1◊3}2}1,2} = {1{1{1◊1,,2}3,,2}2}
{c 1{2`}1{1{1◊1,2}2}2} = {1{1◊1{1◊1,,2}1,2,,2}2}
{c 1{2`}1{1{1◊1,2}2}1,2} = {1{1◊1{1◊1,,2}1{1◊1,,2}2,,2}2}
{c 1{2`}1{1{1◊1`2}2}2} = {1{1◊1{1,,2◊1,,2}2,,2}2}
{c 1{2`}1{1{1◊1{1{2,,}2}2}2}2} = {1{1◊1{1{2,,}2◊1,,2}2,,2}2}
{c 1{2`}1{c 1{2`}1,3}2} = {1{1◊1{1◊2,,2}2,,2}2}
{c 1{2`}1{c 1{2`}1,1,2}2} = {1{1◊1,,3}2}
{c 1{2`}1{c 1{2`}1{c 1{2`}1,2}2}2} = {1{1◊2,,3}2}
{c 1{2`}1{c 1{2`}1{c 1{2`}1{c 1{2`}1,2}2}2}2} = {1{1◊2,,4}2}
{c 1{2`}1`2} = {1{1◊1,,1,2}2}
{c 2{2`}1`2} = {1{1◊1,,1{1,,1,2}2}2}
{c 1`2{2`}1`2} = {1{1◊1,,1{1,,1,,1,2}2}2}
{c 1{2`}2`2} = {1{1◊1,,1{1{2,,}2}2}2}
{c 1{2`}1,2`2} = {1{1◊1,,1{1◊2}2}2}
{c 1{2`}1{c 1{2`}1`2}2`2} = {1{1◊1,,1{1◊1{1◊1,,1,2}2}2}2}
{c 1{2`}1{c 1{2`}1`2}1,2`2} = {1{1◊1,,1{1◊1,,2}2}2}
{c 1{2`}1{1{1,,2◊1,,1,2}2}2`2} = {1{1,,2◊1,,1{1◊1,,2}2}2}
{c 1{2`}1{1{1◊2,,1,2}2}2`2} = {1{1◊2,,1{1◊1,,2}2}2}
{c 1{2`}1{1{1◊1,,2,2}2}2`2} = {1{1◊1,,2{1◊1,,2}2}2}
{c 1{2`}1{1{1◊1,,1,3}2}2`2} = {1{1◊1,,1,2{1◊1,,2}2}2}
{c 1{2`}1{1{1◊1,,1,3}2}1,2`2} = {1{1◊1,,1{1◊1,,2}3}2}
{c 1{2`}1{1{1◊1,,1`2}2}2`2} = {1{1◊1,,1{1,,2◊1,,2}2}2}
{c 1{2`}1{c 1{2`}1{c 1{2`}1`2}1,2`2}2`2} = {1{1◊1,,1{1◊1,,3}2}2}
{c 1{2`}1`3} = {1{1◊1,,1{1◊1,,1,2}2}2}
{c 1{2`}1`4} = {1{1◊1,,1{1◊1,,1{1◊1,,1,2}2}2}2}
{c 1{2`}1`1,2} = {1{1◊1,,1,,2}2}
{c 1{2`}1{c 1{2`}1`1,2}1,2`1,2} = {1{1◊1,,1{1◊1,,1,,2}1{1◊1,,2}2}2}
{c 1{2`}1{1{1◊2,,1{1◊1,,1,,2}2}2}2`1,2} = {1{1◊2,,1{1◊1,,1,,2}1{1◊1,,2}2}2}
{c 1{2`}1{1{1◊1,,2{1◊1,,1,,2}2}2}2`1,2} = {1{1◊1,,2{1◊1,,1,,2}1{1◊1,,2}2}2}
{c 1{2`}1{1{1◊1,,1{1◊1,,1,,2}3}2}2`1,2} = {1{1◊1,,1{1◊1,,1,,2}2{1◊1,,2}2}2}
{c 1{2`}1{1{1◊1,,1{1◊1,,1,,2}3}2}1,2`1,2} = {1{1◊1,,1{1◊1,,1,,2}1{1◊1,,2}3}2}
{c 1{2`}1{1{1◊1,,1{1◊1,,1,,2}1`2}2}2`1,2} = {1{1◊1,,1{1◊1,,1,,2}1{1,,2◊1,,2}2}2}
{c 1{2`}1{c 1{2`}1{c 1{2`}1`1,2}1,2`1,2}2`1,2} = {1{1◊1,,1{1◊1,,1,,2}1{1◊1,,3}2}2}
{c 1{2`}1`2,2} = {1{1◊1,,1{1◊1,,1,,2}1{1◊1,,1,2}2}2}
{c 1{2`}1`1,3} = {1{1◊1,,1{1◊1,,1,,2}1{1◊1,,1{1◊1,,1,,2}2}2}2}
{c 1{2`}1`1,1,2} = {1{1◊1,,1{1◊1,,1,,2}1{1◊1,,1,,2}2}2}
{c 1{2`}1`1{2}2} = {1{1◊1,,1{2◊1,,1,,2}2}2}
{c 1{2`}1`1{1`2}2} = {1{1,,2◊1,,1,,2}2}
{c 1{2`}1`1{1{1{2,,}2}2}2} = {1{1{2,,}2◊1,,1,,2}2}
{c 1{2`}1`1{1{1{2,,}2}2}1,2} = {1{1{1{1◊1,,1,,2}2,,}2◊1,,1,,2}2}
{c 1{2`}1`1{1{1{1,,2,,}2}2}2} = {1{1◊2,,1,,2}2}
{c 1{2`}1`1{1{1◊1,,2}2}2} = {1{1◊1,,2,,2}2}
{c 1{2`}1`1{c 1{2`}1`1,2}2} = {1{1◊1,,1,,3}2}
{c 1{2`}1`1{c 1{2`}1`1{c 1{2`}1`1,2}2}2} = {1{1◊1,,1,,4}2}
{c 1{2`}1`1`2} = {1{1◊1,,1,,1,2}2}
{c 1{2`}1`1`1`2} = {1{1◊1,,1,,1,,1,2}2}
{c 1{2`}1{2`}2} = {1{1◊1{2,,}2}2}
{c 1{2`}1{2`}1,2} = {1{1◊1◊2}2}
{c 1{2`}1{2`}1`2} = {1{1◊1◊1,,1,2}2}
{c 1{2`}1{2`}1{2`}2} = {1{1◊1◊1{2,,}2}2}
{c 1{3`}2} = {1{1{2{1{1,,2,,}2}2,,}2}2}
{c 2{3`}2} = {1{1{1{1,,1,2}2{1{1,,2,,}2}2,,}2}2}
{c 1{2`}2{3`}2} = {1{1{1{1{2,,}2}2{1{1,,2,,}2}2,,}2}2}
{c 1{2`}1,2{3`}2} = {1{1{1{1◊2}2{1{1,,2,,}2}2,,}2}2}
{c 1{2`}1`2{3`}2} = {1{1{1{1◊1,,1,2}2{1{1,,2,,}2}2,,}2}2}
{c 1{2`}1{2`}2{3`}2} = {1{1{1{1◊1{2,,}2}2{1{1,,2,,}2}2,,}2}2}
{c 1{3`}3} = {1{1{1{1{2{1{1,,2,,}2}2,,}2}2{1{1,,2,,}2}2,,}2}2}
{c 1{3`}1,2} = {1{1{1{1{1,,2,,}2}3,,}2}2}
{c 1{3`}1,1,2} = {1{1{1{1{1,,2,,}2}3,,}1,,2}2}
{c 1{3`}1`2} = {1{1{1{1{1,,2,,}2}3,,}1,,1,2}2}
{c 1{3`}1{2`}2} = {1{1{1{1{1,,2,,}2}3,,}1{2,,}2}2}
{c 1{3`}1{2`}1,2} = {1{1{1{1{1,,2,,}2}3,,}1{1{1{1,,2,,}2}2,,}2}2}
{c 1{3`}1{3`}2} = {1{1{1{1{1,,2,,}2}3,,}1{2{1{1,,2,,}2}2,,}2}2}
{c 1{3`}1{3`}1,2} = {1{1{1{1{1,,2,,}2}3,,}1{1{1{1,,2,,}2}3,,}2}2}
{c 1{4`}2} = {1{1{2{1{1,,2,,}2}3,,}2}2}
{c 1{5`}2} = {1{1{2{1{1,,2,,}2}4,,}2}2}
{c 1{1,2`}2} = {1{1{1{1{1,,2,,}2}1,2,,}2}2}
{c 1{1,2`}1,2} = {1{1{1{1{1,,2,,}2}1{1{1,,2,,}2}2,,}2}2}
{c 1{1,2`}1,1,2} = {1{1{1{1{1,,2,,}2}1{1{1,,2,,}2}2,,}1,,2}2}
{c 1{1,2`}1`2} = {1{1{1{1{1,,2,,}2}1{1{1,,2,,}2}2,,}1,,1,2}2}
{c 1{1,2`}1{2`}2} = {1{1{1{1{1,,2,,}2}1{1{1,,2,,}2}2,,}1{2,,}2}2}
{c 1{1,2`}1{1,2`}2} = {1{1{1{1{1,,2,,}2}1{1{1,,2,,}2}2,,}1{1{1{1,,2,,}2}1,2,,}2}2}
{c 1{2,2`}2} = {1{1{2{1{1,,2,,}2}1{1{1,,2,,}2}2,,}2}2}
{c 1{1,3`}2} = {1{1{1{1{1,,2,,}2}1,2{1{1,,2,,}2}2,,}2}2}
{c 1{1,3`}1,2} = {1{1{1{1{1,,2,,}2}1{1{1,,2,,}2}3,,}2}2}
{c 1{1,1,2`}1,2} = {1{1{1{1{1,,2,,}2}1{1{1,,2,,}2}1,2,,}2}2}
{c 1{1,1,1,2`}2} = {1{1{1{1{1,,2,,}2}1{1{1,,2,,}2}1{1{1,,2,,}2}1,2,,}2}2}
{c 1{1{2}2`}2} = {1{1{1{2{1,,2,,}2}2,,}2}2}
{c 1{1{2}2`}1,2} = {1{1{1{1,,2,,}2}2{1,,2,,}2}2}
{c 1{1{1`2}2`}2} = {1{1,,2{1,,2,,}2}2}
{c 1{1{c 1}2`}2} = {1{1,,1,2{1,,2,,}2}2}
{c 1{1{c 1{2`}2}2`}2} = {1{1{2,,}2{1,,2,,}2}2}
{c 1{1{c 1{2`}2}2`}1,2} = {1{1{1{1{1,,2,,}2}2,,}2{1,,2,,}2}2}
{c 1{1{c 1{2`}1,2}2`}2} = {1{1{1,,2,,}3}2}
{c 1{1{c 1{1{c 1{2`}2}2`}2}2`}2} = {1{1{2,,}2{1,,2,,}3}2}
{c 1{1{c 1{1{c 1{1{c 1{2`}2}2`}2}2`}2}2`}2} = {1{1{2,,}2{1,,2,,}4}2}
{c 1{1`2`}2} = {1{1{1,,2,,}1,2}2}
{c 2{1`2`}2} = {1{1{1,,2,,}1{1,,1,2}2}2}
{c 1`2{1`2`}2} = {1{1{1,,2,,}1{1,,1,,1,2}2}2}
{c 1{2`}2{1`2`}2} = {1{1{1,,2,,}1{1{2,,}2}2}2}
{c 1{2`}1,2{1`2`}2} = {1{1{1,,2,,}1{1{1{1{1,,2,,}2}2,,}2}2}2}
{c 1{3`}2{1`2`}2} = {1{1{1,,2,,}1{1{2{1{1,,2,,}2}2,,}2}2}2}
{c 1{1{1`2}2`}2{1`2`}2} = {1{1{1,,2,,}1{1{1{1,,2{1,,2,,}2}2,,}2}2}2}
{c 1{1{c 1{1`2`}2}2`}2{1`2`}2} = {1{1{1,,2,,}1{1{1{1{1,,2,,}1,2}2,,}2}2}2}
{c 1{1{c 1{1`2`}2}2`}3{1`2`}2} = {1{1{1,,2,,}1{1{1{1{1,,2,,}1{1{1{1{1,,2,,}1,2}2,,}2}2}2,,}2}2}2}
{c 1{1{c 1{1`2`}2}2`}1,2{1`2`}2} = {1{1{1,,2,,}1{1{1,,2,,}2}2}2}
{c 1{1{c 1{1`2`}2}2`}1,1,2{1`2`}2} = {1{1{1{1{1,,2,,}1{1{1,,2,,}2}2}2,,}1,,2}2}
{c 1{1{c 1{1`2`}2}2`}1{1{1{2,,}2}2}2{1`2`}2} = {1{1{2,,}2{1{1{1,,2,,}1{1{1,,2,,}2}2}2,,}1,,2}2}
{c 1{1{c 1{1`2`}2}2`}1{1{1{1{1{1,,2,,}1,,1,2}2,,}2}2}2{1`2`}2} = {1{1{1{1{1,,2,,}1,,1,2}2,,}2{1{1{1,,2,,}1{1{1,,2,,}2}2}2,,}1,,2}2}

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Necessary of the Rule 4

Due to a further problem from this, the Rule 4 (shown below) is necessary.

  • Rule 4: {# A 1 B #′} = {# B #′} if lv(A) < lv(B)

The problem first comes with pDAN. In mEAN, the double grave accent reduces to {1` `2`}, then expands to {1{1…{1{1,2` `1`}2` `1`}…2` `1`}2` `1`}. Notice that the tailing rule apply in this case, so it becomes {1{1…{1{1,2`}2`}…2`}2`}. However, in pDAN without the Rule 4, {1,,3} reduces to {1{1,,3}2,,2}, then expands to {1…{1 {1,2{1,,3}1,,2} 2{1,,3}1,,2}…2{1,,3}1,,2}. For a lower {1{1,,3}1,,2}, we meet the blue double comma, then it reduces to {1{1,,3}1{1{1,,3}1,,2}2}, then {1{1,,3}1…{1{1,,3}1 {1{1,,3}1,2} 2}…2}. And a lower {1{1,,3}2} can reduce to {1{1{1,,3}2,,2}2}, and now we loop back to the first step. So {1,,3} never solves in pDAN.

In {# {1,,3} 1 ,, 2}, the {1,,3} is “blocked” by the double comma, and it’s not the last separator in {# {1,,3} 1 ,, 2}, so the tailing rule doesn’t apply on it. That’s the reason of the problem. To fix this, we enable the Rule 4, then {# {1,,3} 1 ,, 2} reduces to {# ,, 2}, and the problem will clear.

Enabling the Rule 4 from pDAN to DAN (where we must enable the Rule 4) making them inconsistent with lower parts from exAN to mEAN. To make the whole “first generation” consistent, we need to enable the Rule 4 from exAN on, so exAN, EAN and mEAN also need the “level comparison” part.

Catching scale

Catching function is an enumeration of the ordinals such that fast-growing hierarchy (FGH) and slow-growing hierarchy (SGH) have similar growth rate (where SGH catch up FGH). Then, catching scale is an enumeration of separators such that fast-growing version and slow-growing version of array notation have similar growth rate. To use it, we first need to define fast-growing and slow-growing versions of array notation.

Slow-growing array notation

The recursion rule in array notation is:

  • Rule 2: s(a,b,c #) = s(a,s(a,b-1,c #),c-1 #)

So the ruleset make it FGH-like. To make an SGH-like array notation, we take

  • Rule 2ss: ss(a,b,c #) = a^ss(a,b,c-1 #)

The other rules and the process remain the same. So ss(a,b,2) = a^a^b, ss(a,b,3) = a^a^a^b, ss(a,b,1,2) = a↑↑(b+1).

The slow version of array notation (ss) seems to grow slow, except at the catching points where ss and s have similar growth rate. But what’s “similar growth rate”?

Similar growth rate

Here is a definition of similar growth rate. For function f and g (on positive integers), f >* g if for all k, there exists such N that for all n > N, f(n) > gk(n) = g(g(…g(g(n))…)) with k g′s. We say f and g have similar growth rate (denoted f ≈ g) if neither f >* g nor g >* f.

In array notation, ss(a,b #) ≤ s(a,b #) for the same a, b and #. So for f(n) = s(n,n #) and g(n) = ss(n,n #), g >* f never happens. So when we say f = λn.s(n,n #) (using the notation of λ-abstraction) and g = λn.ss(n,n #) have similar growth rate, it’s (not f >* g), i.e. there exists such k that there exist arbitrarily large n that gk(n) ≥ f(n).

Catching scale

Finally, let {c 1} = max{A∈S|¬∃B∈S(λn.s(n,n A 2) >* λn.s(n,n B 2))} where the max is by level comparison, and S = {A is separator|λn.s(n,n A 2) ≈ λn.ss(n,n A 2)}. And {c m #} (m > 1) is defined as {c m #} = max{A∈S|λn.s(n,n A 2) >* λn.s(n,n {c m-1 #} 2)∧ ¬∃B∈S(λn.s(n,n A 2) >* λn.s(n,n B 2) >* λn.s(n,n {c m-1 #} 2))}. The symbol c means “catching” here.

Generally, {c 1} is the first “catching separator” A such that λn.s(n,n A 2) ≈ λn.ss(n,n A 2) (i.e. slow-growing version of array notation catches up the normal one), and {c m #} is the next catching separator after {c m-1 #}. Here the “first” and “next” are based on growth rate of s(n,n A 2). But what does the “max by level comparison” means? There are different separators A such that s(n,n A 2) have similar growth rates, but not identical to each other. And this “max by level comparison” pick the separator with highest level.

Here is comparisons between ss and s.

The comma in ss corresponds to s(n,s(n,n))
ss(n,n,3) corresponds to s(n,s(n,s(n,n)))
ss(n,n,1,2) corresponds to s(n,n+1,2)
ss(n,n,2,2) corresponds to s(n,n+2,2)
ss(n,n,1,3) corresponds to s(n,2n,2)
ss(n,n,1,4) corresponds to s(n,3n-1,2)
ss(n,n,1,1,2) corresponds to s(n,n^2,2)
ss(n,n,1,1,1,2) corresponds to s(n,n^3,2)
ss(n,n,1,1,1,1,2) corresponds to s(n,n^4,2)
{2} in ss corresponds to s(n,s(n,n),2)
ss(n,n,2{2}2) corresponds to s(n,s(n,n)+1,2)
ss(n,n,1,2{2}2) corresponds to s(n,s(n,n)+n,2)
ss(n,n,1,1,2{2}2) corresponds to s(n,s(n,n)+n^2,2)
ss(n,n{2}3) corresponds to s(n,s(n,n)×2,2)
ss(n,n{2}1,2) corresponds to s(n,s(n,n+1),2)
ss(n,n{2}2,2) corresponds to s(n,s(n,n+1)+s(n,n),2)
ss(n,n{2}1,3) corresponds to s(n,s(n,n+1)×2,2)
ss(n,n{2}1,1,2) corresponds to s(n,s(n,n+2),2)
ss(n,n{2}1{2}2) corresponds to s(n,s(n,2n),2)
ss(n,n{2}1{2}1{2}2) corresponds to s(n,s(n,3n),2)
{3} in ss corresponds to s(n,s(n,n^2),2)
ss(n,n{3}3) corresponds to s(n,s(n,n^2)×2,2)
ss(n,n{3}1,2) corresponds to s(n,s(n,n^2+1),2)
ss(n,n{3}1{2}2) corresponds to s(n,s(n,n^2+n),2)
ss(n,n{3}1{3}2) corresponds to s(n,s(n,n^2×2),2)
{4} in ss corresponds to s(n,s(n,n^3),2)
{1,2} in ss corresponds to s(n,s(n,s(n,n)),2)
ss(n,n{1,2}1,2) corresponds to s(n,s(n,s(n,n)+1),2)
{2,2} in ss corresponds to s(n,s(n,s(n,n+1)),2)
{1,3} in ss corresponds to s(n,s(n,s(n,2n)),2)
{1,1,2} in ss corresponds to s(n,s(n,s(n,n^2)),2)
{2,1,2} in ss corresponds to s(n,s(n,s(n,n^2+1)),2)
{1,2,2} in ss corresponds to s(n,s(n,s(n,n^2+n)),2)
{1,1,3} in ss corresponds to s(n,s(n,s(n,n^2×2)),2)
{1,1,1,2} in ss corresponds to s(n,s(n,s(n,n^3)),2)
{1{2}2} in ss corresponds to s(n,s(n,s(n,s(n,n))),2)
{2{2}2} in ss corresponds to s(n,s(n,s(n,s(n,n)+1)),2)
{1{2}3} in ss corresponds to s(n,s(n,s(n,s(n,n)×2)),2)
{1{2}1,2} in ss corresponds to s(n,s(n,s(n,s(n,n+1))),2)
{1{2}1{2}2} in ss corresponds to s(n,s(n,s(n,s(n,2n))),2)
{1{3}2} in ss corresponds to s(n,s(n,s(n,s(n,n^2))),2)
{1{4}2} in ss corresponds to s(n,s(n,s(n,s(n,n^3))),2)
{1{1,2}2} in ss corresponds to s(n,s(n,s(n,s(n,s(n,n)))),2)
{2{1,2}2} in ss corresponds to s(n,s(n,s(n,s(n,s(n,n))+1)),2)
{1{1,2}1,2} in ss corresponds to s(n,s(n,s(n,s(n,s(n,n)+1))),2)
{1{2,2}2} in ss corresponds to s(n,s(n,s(n,s(n,s(n,n+1)))),2)
{1{1{2}2}2} in ss corresponds to s(n,s(n,s(n,s(n,s(n,s(n,n))))),2)
{1{1{1,2}2}2} in ss corresponds to s(n,s(n,s(n,s(n,s(n,s(n,s(n,n)))))),2)
{1{1{1{2}2}2}2} in ss corresponds to s(n,s(n,s(n,s(n,s(n,s(n,s(n,s(n,n))))))),2)
{1`2} in ss corresponds to s(n,s(n,n,2),2)
{2`2} in ss corresponds to s(n,s(n,s(n,s(n,n-2,2)+1)),2)
{1{1`2}2`2} in ss corresponds to s(n,s(n,s(n,s(n,n,2)×2)),2)
{1{1`2}1{1`2}2`2} in ss corresponds to s(n,s(n,n+4,2),2)
{1{1{1`2}1{1`2}2`2}2`2} in ss corresponds to s(n,s(n,n+6,2),2)
{1`3} in ss corresponds to s(n,s(n,2n,2),2)
{1`4} in ss corresponds to s(n,s(n,3n,2),2)
{1`1,2} in ss corresponds to s(n,s(n,n^2,2),2)
{1`2,2} in ss corresponds to s(n,s(n,n^2+n,2),2)
{1`1,3} in ss corresponds to s(n,s(n,n^2×2,2),2)
{1`1,1,2} in ss corresponds to s(n,s(n,n^3,2),2)
{1`1{2}2} in ss corresponds to s(n,s(n,s(n,n),2),2)
{1`1{1,2}2} in ss corresponds to s(n,s(n,s(n,s(n,n)),2),2)
{1`1{1`2}2} in ss corresponds to s(n,s(n,s(n,n,2),2),2)
{1`1{1`3}2} in ss corresponds to s(n,s(n,s(n,n^2,2),2),2)
{1`1{1`1{1`2}2}2} in ss corresponds to s(n,s(n,s(n,s(n,n,2),2),2),2)
{1`1`2} in ss corresponds to s(n,n,3)
{1`2`2} in ss corresponds to s(n,s(n,s(n,n-2,3)+n,2),2)
{1`1{1`1`2}2`2} in ss corresponds to s(n,s(n,s(n,n,3),2),2)
{1`1{1`2`2}2`2} in ss corresponds to s(n,s(n,s(n,s(n,n-1,3)+n,2),2),2)
{1`1{1`1{1`1`2}2`2}2`2} in ss corresponds to s(n,s(n,s(n,s(n,n,3),2),2),2)
{1`1`3} in ss corresponds to s(n,2n,3)
{1`1`1,2} in ss corresponds to s(n,n^2,3)
{1`1`1{2}2} in ss corresponds to s(n,n^n,3)
{1`1`1{1`1`2}2} in ss corresponds to s(n,s(n,n,3),3)
{1`1`1`2} in ss corresponds to s(n,n,4)
{1`1`1`1`2} in ss corresponds to s(n,n,5)
{1{2`}2} in ss corresponds to s(n,n,1,2)
{1`2{2`}2} in ss corresponds to s(n,s(n,s(n,n,n)+n,2),2)
{1`1{1`2{2`}2}2{2`}2} in ss corresponds to s(n,s(n,s(n,s(n,n,n)+n,2),2),2)
{1`1`2{2`}2} in ss corresponds to s(n,s(n,n,n)+n,3)
{1`1`1`2{2`}2} in ss corresponds to s(n,s(n,n,n)+n,4)
{1{2`}3} in ss corresponds to s(n,s(n,n,n),n)
{1{2`}4} in ss corresponds to s(n,s(n,s(n,n,n),n),n)
{1{2`}1,2} in ss corresponds to s(n,n+1,1,2)
{1{2`}2,2} in ss corresponds to s(n,n+1,n+1)
{1{2`}1,3} in ss corresponds to s(n,2n,n+1)
{1{2`}1,1,2} in ss corresponds to s(n,n^2,n+1)
{1{2`}1{2}2} in ss corresponds to s(n,s(n,n),n+1)
{1{2`}1{1{2`}1,2}2} in ss corresponds to s(n,s(n,n,n+1),n+1)
{1{2`}1`2} in ss corresponds to s(n,n+2,1,2)
{1{2`}1`1`2} in ss corresponds to s(n,n+3,1,2)
{1{2`}1{2`}2} in ss corresponds to s(n,2n,1,2)
{1{3`}2} in ss corresponds to s(n,n^2,1,2)
{1{1,2`}2} in ss corresponds to s(n,s(n,n),1,2)
{1{1{1{1,2`}2}2`}2} in ss corresponds to s(n,s(n,s(n,n),1,2),1,2)
{1{1`2`}2} in ss corresponds to s(n,n,2,2)
{1`2{1`2`}2} in ss corresponds to s(n,s(n,s(n,n,2,2)+n,2),2)
{1`1`2{1`2`}2} in ss corresponds to s(n,s(n,n,2,2)+n,3)
{1{2`}2{1`2`}2} in ss corresponds to s(n,s(n,n,2,2),n)
{1{1,2`}2{1`2`}2} in ss corresponds to s(n,s(n,n,2,2),s(n,n))
{1{1{1{1`2`}2}2`}2{1`2`}2} in ss corresponds to s(n,n+1,2,2)
{1{1,2`}2{1{1{1`2`}2}2`}2{1`2`}2} in ss corresponds to s(n,s(n,n+1,2,2),s(n,n))
{1{1{1{1`2`}2}2`}3{1`2`}2} in ss corresponds to s(n,s(n,n+1,2,2),s(n,n,1,2))
{1{1{1`2{1`2`}2}2`}2{1`2`}2} in ss corresponds to s(n,s(n,n+1,2,2),s(n,s(n,n,1,2)+n,2))
{1{1{1{1{1{1`2`}2}2`}2{1`2`}2}2`}2{1`2`}2} in ss corresponds to s(n,n+2,2,2)
{1{1`2`}3} in ss corresponds to s(n,2n,2,2)
{1{1`2`}1,2} in ss corresponds to s(n,n^2,2,2)
{1{1`2`}1`2} in ss corresponds to s(n,n,3,2)
{1{1`2`}1{2`}2} in ss corresponds to s(n,n,1,3)
{1{1`2`}1{1`2`}2} in ss corresponds to s(n,n,2,3)
{1{1`2`}1{1`2`}1{1`2`}2} in ss corresponds to s(n,n,2,4)
{1{2`2`}2} in ss corresponds to s(n,n,1,1,2)
{1{1,2`2`}2} in ss corresponds to s(n,s(n,n),1,1,2)
{1{1`3`}2} in ss corresponds to s(n,n,2,1,2)
{1{1{1{1`3`}2}2`}2{1`3`}2} in ss corresponds to s(n,n+1,2,1,2)
{1{1`3`}3} in ss corresponds to s(n,2n,2,1,2)
{1{1`3`}1,2} in ss corresponds to s(n,n^2,2,1,2)
{1{1`3`}1`2} in ss corresponds to s(n,n,3,1,2)
{1{1`3`}1{2`}2} in ss corresponds to s(n,n,1,2,2)
{1{1`3`}1{1`2`}2} in ss corresponds to s(n,n,2,2,2)
{1{1`3`}1{1`2`}1{1`2`}2} in ss corresponds to s(n,n,2,3,2)
{1{1`3`}1{2`2`}2} in ss corresponds to s(n,n,1,1,3)
{1{1`3`}1{1`3`}2} in ss corresponds to s(n,n,2,1,3)
{1{1`3`}1{1`3`}1{1`3`}2} in ss corresponds to s(n,n,2,1,4)
{1{2`3`}2} in ss corresponds to s(n,n,1,1,1,2)
{1{1`4`}2} in ss corresponds to s(n,n,2,1,1,2)
{1{1`5`}2} in ss corresponds to s(n,n,2,1,1,1,2)
{1{1`1,2`}2} in ss corresponds to s(n,n{2}2)
{1{1`1`2`}2} in ss corresponds to s(n,n,2{2}2)
{1{1`1`2`}1`2} in ss corresponds to s(n,n,3{2}2)
{1{1`1`2`}1{1`2`}2} in ss corresponds to s(n,n,2,2{2}2)
{1{1`1`2`}1{1`3`}2} in ss corresponds to s(n,n,2,1,2{2}2)
{1{1`1`2`}1{1`1,2`}2} in ss corresponds to s(n,n{2}3)
{1{1`1`2`}1{1`1`2`}2} in ss corresponds to s(n,n,2{2}3)
{1{2`1`2`}2} in ss corresponds to s(n,n{2}1,2)
{1{1`2`2`}2} in ss corresponds to s(n,n,2{2}1,2)
{1{1`3`2`}2} in ss corresponds to s(n,n,2{2}1,1,2)
{1{1`1`3`}2} in ss corresponds to s(n,n,2{2}1{2}2)
{1{1`1`1`2`}2} in ss corresponds to s(n,n,2{3}2)
{1{1{2`}2`}2} in ss corresponds to s(n,n{1,2}2)
{1{1{1`2`}2`}2} in ss corresponds to s(n,n,2{1,2}2)
{1{1{1`1,2`}2`}2} in ss corresponds to s(n,n{1{2}2}2)
{1{1{1`1`2`}2`}2} in ss corresponds to s(n,n,2{1{2}2}2)
{1{1{1{2`}2`}2`}2} in ss corresponds to s(n,n{1{1,2}2}2)
{1{1` `2`}2} in ss corresponds to s(n,n{1`2}2)
{1{1{1` ` `2` `}2`}2} in ss corresponds to s(n,n{1{1` `2`}2}2)
{1{1{1{1` ` ` `2` ` `}2` `}2`}2} in ss corresponds to s(n,n{1{1{1` ` `2` `}2`}2}2)

Generally, the {#` `…` `} with m grave accents in ss approximately corresponds to {#` `…`} with m-1 grave accents in s. So {c 1} = {1,,1,2}.

And here are more values of catching scale.

{c 2} = {1,,1{1,,1,2}2}
{c 3} = {1,,1{1,,1{1,,1,2}2}2}
{c 1,2} = {1,,1,,2} (the process still remain the same)
{c 2,2} = {1,,1{1,,1,,2}1{1,,1,2}2}
{c 3,2} = {1,,1{1,,1,,2}1{1,,1{1,,1,2}2}2}
{c 1,3} = {1,,1{1,,1,,2}1{1,,1{1,,1,,2}2}2}
{c 2,3} = {1,,1{1,,1,,2}1{1,,1{1,,1,,2}1{1,,1,2}2}2}
{c 1,4} = {1,,1{1,,1,,2}1{1,,1{1,,1,,2}1{1,,1{1,,1,,2}2}2}2}
{c 1,1,2} = {1,,1{1,,1,,2}1{1,,1,,2}2}
{c 2,1,2} = {1,,1{1,,1,,2}1{1,,1,,2}1{1,,1,2}2}
{c 1,2,2} = {1,,1{1,,1,,2}1{1,,1,,2}1{1,,1{1,,1,,2}2}2}
{c 1,1,3} = {1,,1{1,,1,,2}1{1,,1,,2}1{1,,1{1,,1,,2}1{1,,1,,2}2}2}
{c 1,1,1,2} = {1,,1{1,,1,,2}1{1,,1,,2}1{1,,1,,2}2}
{c 1,1,1,1,2} = {1,,1{1,,1,,2}1{1,,1,,2}1{1,,1,,2}1{1,,1,,2}2}
{c 1{2}2} = {1,,1{2,,1,,2}2}
{c 2{2}2} = {1,,1{1{1,,1,2}2,,1,,2}2}
{c 1{2}3} = {1,,1{1{1,,1{2,,1,,2}2}2,,1,,2}2}
{c 1{2}1,2} = {1{1,,1,,2}2,,1,,2}
{c 2{2}1,2} = {1,,1{1{1,,1,,2}2,,1,,2}1{1,,1,2}2}
{c 1{2}1,1,2} = {1,,1{1{1,,1,,2}2,,1,,2}1{1,,1,,2}2}
{c 1{2}1,1,1,2} = {1,,1{1{1,,1,,2}2,,1,,2}1{1,,1,,2}1{1,,1,,2}2}
{c 1{2}1{2}2} = {1,,1{1{1,,1,,2}2,,1,,2}1{2,,1,,2}2}
{c 1{2}1{2}1,2} = {1,,1{1{1,,1,,2}2,,1,,2}1{1{1,,1,,2}2,,1,,2}2}
{c 1{2}1{2}1{2}2} = {1,,1{1{1,,1,,2}2,,1,,2}1{1{1,,1,,2}2,,1,,2}1{2,,1,,2}2}
{c 1{3}2} = {1,,1{2{1,,1,,2}2,,1,,2}2}
{c 1{3}1,2} = {1{1,,1,,2}3,,1,,2}
{c 2{3}1,2} = {1,,1{1{1,,1,,2}3,,1,,2}1{1,,1,2}2}
{c 1{3}1,1,2} = {1,,1{1{1,,1,,2}3,,1,,2}1{1,,1,,2}2}
{c 1{3}1{2}2} = {1,,1{1{1,,1,,2}3,,1,,2}1{2,,1,,2}2}
{c 1{3}1{3}2} = {1,,1{1{1,,1,,2}3,,1,,2}1{2{1,,1,,2}2,,1,,2}2}
{c 1{4}2} = {1,,1{2{1,,1,,2}3,,1,,2}2}
{c 1{5}2} = {1,,1{2{1,,1,,2}4,,1,,2}2}
{c 1{1,2}2} = {1{1,,1,,2}1,2,,1,,2}
{c 2{1,2}2} = {1{1,,1,,2}1{1,,1,2}2,,1,,2}
{c 1{1,2}1,2} = {1{1,,1,,2}1{1,,1,,2}2,,1,,2}
{c 2{1,2}1,2} = {1,,1{1{1,,1,,2}1{1,,1,,2}2,,1,,2}1{1,,1,2}2}
{c 1{1,2}1,1,2} = {1,,1{1{1,,1,,2}1{1,,1,,2}2,,1,,2}1{1,,1,,2}2}
{c 1{1,2}1{2}2} = {1,,1{1{1,,1,,2}1{1,,1,,2}2,,1,,2}1{2,,1,,2}2}
{c 1{1,2}1{3}2} = {1,,1{1{1,,1,,2}1{1,,1,,2}2,,1,,2}1{2{1,,1,,2}2,,1,,2}2}
{c 1{1,2}1{1,2}2} = {1,,1{1{1,,1,,2}1{1,,1,,2}2,,1,,2}1{1{1,,1,,2}1,2,,1,,2}2}
{c 1{1,2}1{1,2}1,2} = {1,,1{1{1,,1,,2}1{1,,1,,2}2,,1,,2}1{1{1,,1,,2}1{1,,1,,2}2,,1,,2}2}
{c 1{2,2}2} = {1,,1{2{1,,1,,2}1{1,,1,,2}2,,1,,2}2}
{c 1{2,2}1,2} = {1{1,,1,,2}2{1,,1,,2}2,,1,,2}
{c 1{3,2}2} = {1,,1{2{1,,1,,2}2{1,,1,,2}2,,1,,2}2}
{c 1{4,2}2} = {1,,1{2{1,,1,,2}3{1,,1,,2}2,,1,,2}2}
{c 1{1,3}2} = {1{1,,1,,2}1,2{1,,1,,2}2,,1,,2}
{c 1{1,3}1,2} = {1{1,,1,,2}1{1,,1,,2}3,,1,,2}
{c 1{1,4}2} = {1{1,,1,,2}1,2{1,,1,,2}3,,1,,2}
{c 1{1,5}2} = {1{1,,1,,2}1,2{1,,1,,2}4,,1,,2}
{c 1{1,1,2}2} = {1{1,,1,,2}1{1,,1,,2}1,2,,1,,2}
{c 1{1,1,2}1,2} = {1{1,,1,,2}1{1,,1,,2}1{1,,1,,2}2,,1,,2}
{c 1{2,1,2}2} = {1,,1{2{1,,1,,2}1{1,,1,,2}1{1,,1,,2}2,,1,,2}2}
{c 1{3,1,2}2} = {1,,1{2{1,,1,,2}2{1,,1,,2}1{1,,1,,2}2,,1,,2}2}
{c 1{1,2,2}2} = {1{1,,1,,2}1,2{1,,1,,2}1{1,,1,,2}2,,1,,2}
{c 1{1,1,3}2} = {1{1,,1,,2}1{1,,1,,2}1,2{1,,1,,2}2,,1,,2}
{c 1{1,1,4}2} = {1{1,,1,,2}1{1,,1,,2}1,2{1,,1,,2}3,,1,,2}
{c 1{1,1,1,2}2} = {1{1,,1,,2}1{1,,1,,2}1{1,,1,,2}1,2,,1,,2}
{c 1{1,1,1,1,2}2} = {1{1,,1,,2}1{1,,1,,2}1{1,,1,,2}1{1,,1,,2}1,2,,1,,2}
{c 1{1{2}2}2} = {1{2,,1,,2}2,,1,,2}
{c 1{1{2}2}1,2} = {1{1{1,,1,,2}2,,1,,2}2,,1,,2}
{c 1{2{2}2}2} = {1,,1{2{1{1,,1,,2}2,,1,,2}2,,1,,2}2}
{c 1{3{2}2}2} = {1,,1{2{1,,1,,2}2{1{1,,1,,2}2,,1,,2}2,,1,,2}2}
{c 1{1,2{2}2}2} = {1{1,,1,,2}1,2{1{1,,1,,2}2,,1,,2}2,,1,,2}
{c 1{1,1,2{2}2}2} = {1{1,,1,,2}1{1,,1,,2}1,2{1{1,,1,,2}2,,1,,2}2,,1,,2}
{c 1{1{2}3}2} = {1{2,,1,,2}2{1{1,,1,,2}2,,1,,2}2,,1,,2}
{c 1{1{2}3}1,2} = {1{1{1,,1,,2}2,,1,,2}3,,1,,2}
{c 1{1{2}1,2}2} = {1{1{1,,1,,2}2,,1,,2}1,2,,1,,2}
{c 1{1{2}1,1,2}2} = {1{1{1,,1,,2}2,,1,,2}1{1,,1,,2}1,2,,1,,2}
{c 1{1{2}1{2}2}2} = {1{1{1,,1,,2}2,,1,,2}1{2,,1,,2}2,,1,,2}
{c 1{1{2}1{2}2}1,2} = {1{1{1,,1,,2}2,,1,,2}1{1{1,,1,,2}2,,1,,2}2,,1,,2}
{c 1{1{3}2}2} = {1{2{1,,1,,2}2,,1,,2}2,,1,,2}
{c 1{1{4}2}2} = {1{2{1,,1,,2}3,,1,,2}2,,1,,2}
{c 1{1{1,2}2}2} = {1{1{1,,1,,2}1,2,,1,,2}2,,1,,2}
{c 1{1{1{2}2}2}2} = {1{1{2,,1,,2}2,,1,,2}2,,1,,2}
{c 1{1{1{1,2}2}2}2} = {1{1{1{1,,1,,2}1,2,,1,,2}2,,1,,2}2,,1,,2}
{c 1{1`2}2} = {1,,2,,2}
{c 2{1`2}2} = {1{1,,2,,2}1{1,,1,2}2,,1,,2}
{c 1{1`2}1,2} = {1{1,,2,,2}1{1,,1,,2}2,,1,,2}
{c 1{2`2}2} = {1,,1{2{1,,2,,2}1{1,,1,,2}2,,1,,2}2}
{c 1{1,2`2}2} = {1{1,,1,,2}1,2{1,,2,,2}1{1,,1,,2}2,,1,,2}
{c 1{1{1,2}2`2}2} = {1{1{1,,1,,2}1,2,,1,,2}2{1,,2,,2}1{1,,1,,2}2,,1,,2}
{c 1{1{1`2}2`2}2} = {1{1{1,,2,,2}2,,1,,2}2{1,,2,,2}1{1,,1,,2}2,,1,,2}
{c 1{1{1`2}2`2}1,2} = {1{1{1,,2,,2}1{1,,1,,2}2,,1,,2}2{1,,2,,2}1{1,,1,,2}2,,1,,2}
{c 1{1{1`2}1,2`2}2} = {1{1{1,,2,,2}1{1,,1,,2}2,,1,,2}1,2{1,,2,,2}1{1,,1,,2}2,,1,,2}
{c 1{1{1`2}1{1`2}2`2}2} = {1{1{1,,2,,2}1{1,,1,,2}2,,1,,2}1{1{1,,2,,2}2,,1,,2}2{1,,2,,2}1{1,,1,,2}2,,1,,2}
{c 1{1{1`2}1{1`2}2`2}1,2} = {1{1{1,,2,,2}1{1,,1,,2}2,,1,,2}1{1{1,,2,,2}1{1,,1,,2}2,,1,,2}2{1,,2,,2}1{1,,1,,2}2,,1,,2}
{c 1{1{2`2}2`2}2} = {1{2{1,,2,,2}1{1,,1,,2}2,,1,,2}2{1,,2,,2}1{1,,1,,2}2,,1,,2}
{c 1{1{1{1`2}2`2}2`2}2} = {1{1{1{1,,2,,2}2,,1,,2}2{1,,2,,2}1{1,,1,,2}2,,1,,2}2{1,,2,,2}1{1,,1,,2}2,,1,,2}
{c 1{1{1{1{1`2}2`2}2`2}2`2}2} = {1{1{1{1{1,,2,,2}2,,1,,2}2{1,,2,,2}1{1,,1,,2}2,,1,,2}2{1,,2,,2}1{1,,1,,2}2,,1,,2}2{1,,2,,2}1{1,,1,,2}2,,1,,2}
{c 1{1`3}2} = {1{1,,2,,2}2{1,,1,,2}2,,1,,2}
{c 1{1`3}1,2} = {1{1,,2,,2}1{1,,1,,2}3,,1,,2}
{c 1{1`1,2}2} = {1{1,,2,,2}1{1,,1,,2}1,2,,1,,2}
{c 1{1`1,1,2}2} = {1{1,,2,,2}1{1,,1,,2}1{1,,1,,2}1,2,,1,,2}
{c 1{1`1{2}2}2} = {1{1,,2,,2}1{2,,1,,2}2,,1,,2}
{c 1{1`1{1,2}2}2} = {1{1,,2,,2}1{1{1,,1,,2}1,2,,1,,2}2,,1,,2}
{c 1{1`1{1{2}2}2}2} = {1{1,,2,,2}1{1{2,,1,,2}2,,1,,2}2,,1,,2}
{c 1{1`1{1`2}2}2} = {1{1,,2,,2}1{1{1,,2,,2}2,,1,,2}2,,1,,2}
{c 1{1`1{1`1{1`2}2}2}2} = {1{1,,2,,2}1{1{1,,2,,2}1{1{1,,2,,2}2,,1,,2}2,,1,,2}2,,1,,2}
{c 1{1`1`2}2} = {1{1,,2,,2}1{1,,2,,2}2,,1,,2}
{c 1{1`1`2}1,2} = {1{1,,2,,2}1{1,,2,,2}1{1,,1,,2}2,,1,,2}
{c 1{1`2`2}2} = {1{1,,2,,2}2{1,,2,,2}1{1,,1,,2}2,,1,,2}
{c 1{1`1,2`2}2} = {1{1,,2,,2}1{1,,1,,2}1,2{1,,2,,2}1{1,,1,,2}2,,1,,2}
{c 1{1`1{1`1`2}2`2}2} = {1{1,,2,,2}1{1{1,,2,,2}1{1,,2,,2}2,,1,,2}2{1,,2,,2}1{1,,1,,2}2,,1,,2}
{c 1{1`1`3}2} = {1{1,,2,,2}1{1,,2,,2}2{1,,1,,2}2,,1,,2}
{c 1{1`1`1,2}2} = {1{1,,2,,2}1{1,,2,,2}1{1,,1,,2}1,2,,1,,2}
{c 1{1`1`1,2}1,2} = {1{1,,2,,2}1{1,,2,,2}1{1,,1,,2}1{1,,1,,2}2,,1,,2}
{c 1{1`2`1,2}2} = {1{1,,2,,2}2{1,,2,,2}1{1,,1,,2}1{1,,1,,2}2,,1,,2}
{c 1{1`1`2,2}2} = {1{1,,2,,2}1{1,,2,,2}2{1,,1,,2}1{1,,1,,2}2,,1,,2}
{c 1{1`1`1{2}2}2} = {1{1,,2,,2}1{1,,2,,2}1{2,,1,,2}2,,1,,2}
{c 1{1`1`1{1`1`1,2}2}2} = {1{1,,2,,2}1{1,,2,,2}1{1{1,,2,,2}1{1,,2,,2}1{1,,1,,2}1,2,,1,,2}2,,1,,2}
{c 1{1`1`1`2}2} = {1{1,,2,,2}1{1,,2,,2}1{1,,2,,2}2,,1,,2}
{c 1{1`1`1`2}1,2} = {1{1,,2,,2}1{1,,2,,2}1{1,,2,,2}1{1,,1,,2}2,,1,,2}
{c 1{1`2`1`2}2} = {1{1,,2,,2}2{1,,2,,2}1{1,,2,,2}1{1,,1,,2}2,,1,,2}
{c 1{1`1`2`2}2} = {1{1,,2,,2}1{1,,2,,2}2{1,,2,,2}1{1,,1,,2}2,,1,,2}
{c 1{1`1`1`3}2} = {1{1,,2,,2}1{1,,2,,2}1{1,,2,,2}2{1,,1,,2}2,,1,,2}
{c 1{1`1`1`1`2}2} = {1{1,,2,,2}1{1,,2,,2}1{1,,2,,2}1{1,,2,,2}2,,1,,2}
{c 1{1{2`}2}2} = {1{2,,2,,2}2,,1,,2}
{c 1{1{1`2`}2}2} = {1{1,,2,,2}2,,2,,2}
{c 1{1{1{1`2`}2`}2}2} = {1{1{1,,2,,2}2,,2,,2}2,,2,,2}
{c 1{1{1` `2`}2}2} = {1,,3,,2}
{c 1{1{1{1` ` `2` `}2`}2}2} = {1,,4,,2}
{c 1{1{1,,1,2}2}2} = {1,,1,2,,2}
{c 1{1{1,,1,2}2}1,2} = {1,,1{1,,1,,2}2,,2}
{c 1{1`2{1,,1,2}2}2} = {1{1,,2,,2}2{1,,1{1,,1,,2}2,,2}2,,1,,2}
{c 1{1{1,,1,2}3}2} = {1{1,,1,2,,2}2{1,,1{1,,1,,2}2,,2}2,,1,,2}
{c 1{1{1,,1,2}1,2}2} = {1{1,,1{1,,1,,2}2,,2}1,2,,1,,2}
{c 1{1{1{1,,1,2}2,,1,2}2}2} = {1{1,,1,2,,2}2{1,,1{1,,1,,2}2,,2}1,,1{1,,1,,2}2,,2}
{c 1{1{1{2,,1,2}2,,1,2}2}2} = {1{2,,1{1,,1,,2}2,,2}2,,1{1,,1,,2}2,,2}
{c 1{1{1,,2,2}2}2} = {1,,2{1,,1,,2}2,,2}
{c 1{1{1,,3,2}2}2} = {1,,3{1,,1,,2}2,,2}
{c 1{1{1,,1,3}2}2} = {1,,1,2{1,,1,,2}2,,2}
{c 1{1{1,,1,3}2}1,2} = {1,,1{1,,1,,2}3,,2}
{c 1{1{1,,1,1,2}2}2} = {1,,1{1,,1,,2}1,2,,2}
{c 1{1{1,,1,1,2}2}1,2} = {1,,1{1,,1,,2}1{1,,1,,2}2,,2}
{c 1{1{1,,1{1`2}2}2}2} = {1,,1{1{1,,2,,2}2,,1,,2}2,,2}
{c 1{1{1,,1{1{1,,1,2}2}2}2}2} = {1,,1{1{1,,1,2,,2}2,,1,,2}2,,2}
{c 1{1{1,,1{1{1,,1{1{1,,1,2}2}2}2}2}2}2} = {1,,1{1{1,,1{1{1,,1,2,,2}2,,1,,2}2,,2}2,,1,,2}2,,2}
{c 1{1{1,,1`2}2}2} = {1,,1{1,,2,,2}2,,2}
{c 1{1{1,,1{1,,3}2}2}2} = {1,,1{1,,3,,2}2,,2}
{c 1{1{1,,1{1,,1,2}2}2}2} = {1,,1{1,,1,2,,2}2,,2}
{c 1{1{1,,1{1,,1{1,,1,2}2}2}2}2} = {1,,1{1,,1{1,,1,2,,2}2,,2}2,,2}
{c 1{1{1,,1,,2}2}2} = {1,,1,,3}
{c 1{1{1,,2{1,,1,,2}2}2}2} = {1,,2{1,,1,,3}2,,2}
{c 1{1{1,,1{1,,1,,2}3}2}2} = {1,,1{1,,1,,3}3,,2}
{c 1{1{1,,1{1,,1,,2}1{1,,1,,2}2}2}2} = {1,,1{1,,1,,3}1{1,,1,,3}2,,2}
{c 1{1{1,,1{2,,1,,2}2}2}2} = {1,,1{2,,1,,3}2,,2}
{c 1{1{1,,2,,2}2}2} = {1,,2,,3}
{c 1{1{1,,3,,2}2}2} = {1,,3,,3}
{c 1{1{1,,1,2,,2}2}2} = {1,,1,2,,3}
{c 1{1{1,,1{1,,1,2,,2}2,,2}2}2} = {1,,1{1,,1,2,,3}2,,3}
{c 1{1{1,,1,,3}2}2} = {1,,1,,4}
{c 1{1{1,,1,,4}2}2} = {1,,1,,5}
{c 1{1{1,,1,,1,2}2}2} = {1,,1,,1,2}

So at {c 1{1{1,,1,,1,2}2}2} the catching scale catches up the “normal” separators.

What about going further? Let’s try {c 1`2}. If we let {c #} with no grave accent at left-superscript of rbrace have lower level than the grave accent, then {c 1`2} expands to {c 1{c 1…{c 1{c 1,2}2}…2}2}. Even if it has reduced to {c 1{c 1}2}, since {c 1} = {1,,1,2}, we still have {c 1{c 1}2} = {c 1{1,,1,2}2}, which will further reduce to {c 1{A,,2}2} with complicated A. Then this process can loop forever, so {c 1`2} never reduces. As a result, we need lv(A) < lv(`) in {c 1 A 2} to make it well-defined. But in that way catching scale will go no faster than “normal separators”, since they have many catching points, e.g. {c 1{1{1,,1,,1{1,,1,,1,2}2}2}2}, {c 1{1{1,,1,,1,,2}2}2}, {c 1{1{1,,2,,1,,2}2}2}, {c 1{1{1,,1,,2,,2}2}2}, {c 1{1{1,,1,,1,,3}2}2}, {c 1{1{1,,1,,1,,1,2}2}2}, etc.

To make a stronger catching scale, the ruleset of the notation shall change, which comes in the next part…

1-separators are a bit stronger than expected

1-separators are actually a bit stronger than in these comparisons.

The problem first appears in {1{1`2}1`2}. It has equal level to {1`2} (though the concept of “level comparison” first came in pDAN), but the recursion level of {1{1`2}1`2} is higher than {1`2}. {1{1`2}1`2} meets the case of grave accent, and expands to {1{1`2}1…{1{1`2}1{1{1`2}1,2}2}…2} (there are some {1`2}’s, which can again expand), while {1`2} only expands to {1…{1{1,2}2}…2}. As a result, {1{1`2}1`2} has recursion level {\varepsilon_1}.

When {1{1`2}2`2} reduces, it first meets the case of “separator neither comma nor grave accent”, where the {1`2} is “pull out”, then becomes {1{1`2}2{1`2}1`2}. The red {1`2} will expand, but the other {1`2} still keeps.

Also, in the subpart “primitive expansion” (using a dot at left-superscript position of rbrace) prior to EAN, {1.1.} reduces to {1.} by the “Rule 2”. However, when the dot leaves the left-superscript position of rbrace, the “Rule 2” won’t apply on it. So {1{1`2}1`2} doesn’t reduce to {1`2} by “Rule 2”.

Generally, {1{1 A n #}1 A n #} has equal level to {1 A n #} (where A is the 1-separator of {1 A n #}), but their recursion level are different. Here are some comparisons showing how this problem makes 1-separators a bit stronger, but more complicated.

  • {1{1`2}1`2} has recursion level {\varepsilon_1}
  • {1{1`2}2`2} has recursion level {\varepsilon_1+\varepsilon_0}
  • {1{1{1`2}1`2}3} has recursion level {\varepsilon_12}
  • {1{2{1`2}1`2}2} has recursion level {\omega^{\omega^{\varepsilon_1+1}}}
  • {1{1{1`2}2`2}2} has recursion level {\omega^{\omega^{\varepsilon_1+\varepsilon_0}}}
  • {1{1{1`2}2`2}1{1{1{1`2}2`2}2}2} has recursion level {\omega^{\omega^{\omega^{\omega^{\varepsilon_1+\varepsilon_0}}}}}
  • {1{1{1`2}2`2}1`2} has recursion level {\varepsilon_2}
  • {1{1{1{1`2}2`2}2`2}1`2} has recursion level {\varepsilon_3}
  • {1`3} has recursion level {\varepsilon_\omega}
  • {1{1`3}3} has recursion level {\varepsilon_\omega2}
  • {1{1`3}1,2} has recursion level {\omega^{\varepsilon_\omega+1}}
  • {1{1`3}1{2}2} has recursion level {\omega^{\varepsilon_\omega+\omega}}
  • {1{1`3}1{1{1`3}2}2} has recursion level {\omega^{\omega^{\varepsilon_\omega2}}}
  • {1{1`3}1`2} has recursion level {\varepsilon_{\omega+1}}
  • {1{1`3}1{1{1`3}2`2}1`2} has recursion level {\varepsilon_{\omega+2}}
  • {1{1`3}1`3} has recursion level {\varepsilon_{\omega2}}
  • {1{1{1`3}2`3}1`3} has recursion level {\varepsilon_{\omega3}}
  • {1`4} has recursion level {\varepsilon_{\omega^2}}
  • {1`5} has recursion level {\varepsilon_{\omega^3}}
  • {1`1,2} has recursion level {\varepsilon_{\omega^\omega}}
  • {1`1 A 2} has recursion level {\varepsilon_{\omega^\alpha}} when {1 A 2} has recursion level α and lv(A) < lv(`)
  • {1`1`2} has recursion level {\varphi(2,0)}
  • {1{1`1`2}1`2} has recursion level {\varepsilon_{\varphi(2,0)+1}}
  • {1{1`1`2}1`1{1{1`1`2}2}2} has recursion level {\varepsilon_{\varphi(2,0)2}}
  • {1{1`1`2}1`1{1{1`1`2}1`2}2} has recursion level {\varepsilon_{\varepsilon_{\varphi(2,0)+1}}}
  • {1{1`1`2}1`1`2} has recursion level {\varphi(2,1)}
  • {1{1{1`1`2}2`1`2}1`1`2} has recursion level {\varphi(2,2)}
  • {1`2`2} has recursion level {\varphi(2,\omega)}
  • {1{1`2`2}1`1`2} has recursion level {\varphi(2,\omega+1)}
  • {1`3`2} has recursion level {\varphi(2,\omega^2)}
  • {1`1,2`2} has recursion level {\varphi(2,\omega^\omega)}
  • {1`1{1`1,2`2}2} has recursion level {\varphi(2,\omega^\omega)}
  • {1`1{1`1,2`2}1`2} has recursion level {\varphi(2,\omega^\omega+1)}
  • {1`2{1`1,2`2}1`2} has recursion level {\varphi(2,\omega^\omega+\omega)}
  • {1`1,2{1`1,2`2}1`2} has recursion level {\varphi(2,\omega^\omega2)}
  • {1`1{1`1,2`2}2`2} has recursion level {\varphi(2,\varphi(2,\omega^\omega))}
  • {1`1{1`1{1`1,2`2}2`2}2`2} has recursion level {\varphi(2,\varphi(2,\varphi(2,\omega^\omega)))}
  • {1`1`3} has recursion level {\varphi(3,0)}
  • {1{1`1`3}1`2} has recursion level {\varepsilon_{\varphi(3,0)+1}}
  • {1{1`1`3}1`1{1{1`1`3}1`2}2} has recursion level {\varepsilon_{\varepsilon_{\varphi(3,0)+1}}}
  • {1{1`1`3}1`1`2} has recursion level {\varphi(2,\varphi(3,0)+1)}
  • {1{1`1`3}1`2`2} has recursion level {\varphi(2,\varphi(3,0)+\omega)}
  • {1{1`1`3}1`1{1{1`1`3}1`1`2}2`2} has recursion level {\varphi(2,\varphi(2,\varphi(3,0)+1))}
  • {1{1`1`3}1`1`3} has recursion level {\varphi(3,1)}
  • {1`2`3} has recursion level {\varphi(3,\omega)}
  • {1`1{1`1`3}2`3} has recursion level {\varphi(3,\varphi(3,0))}
  • {1`1`4} has recursion level {\varphi(4,0)}
  • {1{1`1`4}1`2} has recursion level {\varepsilon_{\varphi(4,0)+1}}
  • {1{1`1`4}1`1`2} has recursion level {\varphi(2,\varphi(4,0)+1)}
  • {1{1`1`4}1`1`3} has recursion level {\varphi(3,\varphi(4,0)+1)}
  • {1{1`1`4}1`1`4} has recursion level {\varphi(4,1)}
  • {1`2`4} has recursion level {\varphi(4,\omega)}
  • {1`1{1`1`4}2`4} has recursion level {\varphi(4,\varphi(4,0))}
  • {1`1`5} has recursion level {\varphi(5,0)}
  • {1`1`1,2} has recursion level {\varphi(\omega,0)}
  • {1{1`1`1,2}1`1`2} has recursion level {\varphi(2,\varphi(\omega,0)+1)}
  • {1{1`1`1,2}1`1`3} has recursion level {\varphi(3,\varphi(\omega,0)+1)}
  • {1{1`1`1,2}1`1`1,2} has recursion level {\varphi(\omega,1)}
  • {1`2`1,2} has recursion level {\varphi(\omega,\omega)}
  • {1`1`2,2} has recursion level {\varphi(\omega+1,0)}
  • {1`1`1,3} has recursion level {\varphi(\omega2,0)}
  • {1`1`1 A 2} has recursion level {\varphi(\alpha,0)} when {1 A 2} has recursion level α and lv(A) < lv(`)
  • {1`1`1`2} has recursion level {\Gamma_0}
  • {1{1`1`1`2}1`2} has recursion level {\varepsilon_{\Gamma_0+1}}
  • {1{1`1`1`2}1`1`2} has recursion level {\varphi(2,\Gamma_0+1)}
  • {1{1`1`1`2}1`1`1,2} has recursion level {\varphi(\omega,\Gamma_0+1)}
  • {1{1`1`1`2}1`1`1{1{1`1`1`2}2}2} has recursion level {\varphi(\Gamma_0,1)}
  • {1{1`1`1`2}1`2`1{1{1`1`1`2}2}2} has recursion level {\varphi(\Gamma_0,\omega)}
  • {1{1`1`1`2}1`1`2{1{1`1`1`2}2}2} has recursion level {\varphi(\Gamma_0+1,0)}
  • {1{1`1`1`2}1`1`1{1{1`1`1`2}1`2}2} has recursion level {\varphi(\varepsilon_{\Gamma_0+1},0)}
  • {1{1`1`1`2}1`1`1{1{1`1`1`2}1`1`1{1{1`1`1`2}2}2}2} has recursion level {\varphi(\varphi(\Gamma_0,1),0)}
  • {1{1`1`1`2}1`1`1`2} has recursion level {\Gamma_1}
  • {1`2`1`2} has recursion level {\Gamma_\omega}
  • {1`1{1`1`1`2}2`1`2} has recursion level {\Gamma_{\Gamma_0}}
  • {1`1`2`2} has recursion level {\alpha\mapsto\Gamma_\alpha} (Gamma-fixed-point)

Here I introduce another ordinal collapsing function, denoted by ψ. Similar to θ, it also uses uncountable cardinals, and is defined as

  • {C_0(\alpha,\beta)=\{\delta|\delta<\beta\}\cup\{0\}}
  • {C_{n+1}(\alpha,\beta)=\{\gamma+\delta,\Omega_{\gamma+1},\psi_\gamma(\eta)|\gamma,\delta,\eta\in C_n(\alpha,\beta)\wedge\eta<\alpha\}}
  • {C(\alpha,\beta)=\bigcup_{n<\omega}C_n(\alpha,\beta)}
  • {\psi_\gamma(\alpha)=\min\{\beta<\Omega_{\gamma+1}|C(\alpha,\beta)\cap\{\delta|\delta<\Omega_{\gamma+1}\}\subseteq\{\delta|\delta<\beta\}\}}

The fundamental sequences can be defined as

  • Co(α12+…+αn)=Co(α1)+Co(α2)+…+Co(αn) where α1 ≥ α2 ≥…≥ αn are powers of ω
  • Co(Ωα+1) = Co(α)+1
  • Co(ψβ(α)) = Co(β)+Co(α) where ψβ(α+1) > ψβ(α)
  • α[n] = max{β<α|Co(β)=Co(α)+n}

And ψ0(α) is shorten as ψ(α). Now the comparisons continue.

  • {1`1`2`2} has recursion level {\theta(\Omega+1,0)=\psi(\Omega^{\Omega+1})}
  • {1`2`2`2} has recursion level {\theta(\Omega+1,\omega)=\psi(\Omega^{\Omega+1}\omega)}
  • {1`1`3`2} has recursion level {\theta(\Omega+2,0)=\psi(\Omega^{\Omega+2})}
  • {1`1`1,2`2} has recursion level {\theta(\Omega+\omega,0)=\psi(\Omega^{\Omega+\omega})}
  • {1`1`1`3} has recursion level {\theta(\Omega2,0)=\psi(\Omega^{\Omega2})}
  • {1`1`1`1,2} has recursion level {\theta(\Omega\omega,0)=\psi(\Omega^{\Omega\omega})}
  • {1`1`1`1`2} has recursion level {\theta(\Omega^2,0)=\psi(\Omega^{\Omega^2})}
  • {1{1`1`1`1`2}1`2} has recursion level {\varepsilon_{\theta(\Omega^2,0)+1}=\psi(\Omega^{\Omega^2}+\Omega)}
  • {1{1`1`1`1`2}1`1`2} has recursion level {\varphi(2,\theta(\Omega^2,0)+1)=\psi(\Omega^{\Omega^2}+\Omega^2)}
  • {1{1`1`1`1`2}1`1`1`2} has recursion level {\Gamma_{\theta(\Omega^2,0)+1}=\psi(\Omega^{\Omega^2}+\Omega^\Omega)}
  • {1{1`1`1`1`2}1`1`1`1{1{1`1`1`1`2}2}2} has recursion level {\theta(\Omega\theta(\Omega^2,0),1)=\psi(\Omega^{\Omega^2}+\Omega^{\Omega\psi(\Omega^{\Omega^2})})}
  • {1{1`1`1`1`2}1`1`1`1`2} has recursion level {\theta(\Omega^2,1)=\psi(\Omega^{\Omega^2}2)}
  • {1`2`1`1`2} has recursion level {\theta(\Omega^2,\omega)=\psi(\Omega^{\Omega^2}\omega)}
  • {1`1`2`1`2} has recursion level {\theta(\Omega^2+1,0)=\psi(\Omega^{\Omega^2+1})}
  • {1`1`1`2`2} has recursion level {\theta(\Omega^2+\Omega,0)=\psi(\Omega^{\Omega^2+\Omega})}
  • {1`1`1`1`3} has recursion level {\theta(\Omega^22,0)=\psi(\Omega^{\Omega^22})}
  • {1`1`1`1`1`2} has recursion level {\theta(\Omega^3,0)=\psi(\Omega^{\Omega^3})}
  • {1{2`}2} has recursion level {\theta(\Omega^\omega,0)=\psi(\Omega^{\Omega^\omega})} (i.e. small Veblen ordinal)
  • {1{1{2`}2}1`2} has recursion level {\varepsilon_{\theta(\Omega^\omega,0)+1}=\psi(\Omega^{\Omega^\omega}+\Omega)}
  • {1{1{2`}2}1`1`1`2} has recursion level {\Gamma_{\theta(\Omega^\omega,0)+1}=\psi(\Omega^{\Omega^\omega}+\Omega^\Omega)}
  • {1{1{2`}2}1`1`1`1`2} has recursion level {\theta(\Omega^2,\theta(\Omega^\omega,0)+1)=\psi(\Omega^{\Omega^\omega}+\Omega^{\Omega^2})}
  • {1{1{2`}2}1{2`}2} has recursion level {\theta(\Omega^\omega,1)=\psi(\Omega^{\Omega^\omega}2)}
  • {1`2{2`}2} has recursion level {\theta(\Omega^\omega,\omega)=\psi(\Omega^{\Omega^\omega}\omega)}
  • {1`1`2{2`}2} has recursion level {\theta(\Omega^\omega+1,0)=\psi(\Omega^{\Omega^\omega+1})}
  • {1`1`1`2{2`}2} has recursion level {\theta(\Omega^\omega+\Omega,0)=\psi(\Omega^{\Omega^\omega+\Omega})}
  • {1`1`1`1`2{2`}2} has recursion level {\theta(\Omega^\omega+\Omega^2,0)=\psi(\Omega^{\Omega^\omega+\Omega^2})}
  • {1{2`}3} has recursion level {\theta(\Omega^\omega2,0)=\psi(\Omega^{\Omega^\omega2})}
  • {1{2`}1,2} has recursion level {\theta(\Omega^\omega\omega,0)=\psi(\Omega^{\Omega^\omega\omega})}
  • {1{2`}1`2} has recursion level {\theta(\Omega^{\omega+1},0)=\psi(\Omega^{\Omega^{\omega+1}})}
  • {1{2`}1{2`}2} has recursion level {\theta(\Omega^{\omega2},0)=\psi(\Omega^{\Omega^{\omega2}})}
  • {1{3`}2} has recursion level {\theta(\Omega^{\omega^2},0)=\psi(\Omega^{\Omega^{\omega^2}})}
  • {1{1,2`}2} has recursion level {\theta(\Omega^{\omega^\omega},0)=\psi(\Omega^{\Omega^{\omega^\omega}})}
  • {1{1{1{1,2`}2}2`}2} has recursion level {\theta(\Omega^{\theta(\Omega^{\omega^\omega},0)},0)=\psi(\Omega^{\Omega^{\psi(\Omega^{\Omega^{\omega^\omega}})}})}
  • {1{1`2`}2} has recursion level {\theta(\Omega^\Omega,0)=\psi(\Omega^{\Omega^\Omega})} (i.e. large Veblen ordinal)
  • {1{1{1`2`}2}1`2} has recursion level {\varepsilon_{\theta(\Omega^\Omega,0)+1}=\psi(\Omega^{\Omega^\Omega}+\Omega)}
  • {1{1{1`2`}2}1`1`1`2} has recursion level {\Gamma_{\theta(\Omega^\Omega,0)+1}=\psi(\Omega^{\Omega^\Omega}+\Omega^\Omega)}
  • {1{1{1`2`}2}1{2`}2} has recursion level {\theta(\Omega^\omega,\theta(\Omega^\Omega,0)+1)=\psi(\Omega^{\Omega^\Omega}+\Omega^{\Omega^\omega})}
  • {1{1{1`2`}2}1{1{1{1{1`2`}2}2}2`}2} has recursion level {\theta(\Omega^{\theta(\Omega^\Omega,0)},1)=\psi(\Omega^{\Omega^\Omega}+\Omega^{\Omega^{\psi(\Omega^{\Omega^\Omega})}})}
  • {1{1{1`2`}2}1{1`2`}2} has recursion level {\theta(\Omega^\Omega,1)=\psi(\Omega^{\Omega^\Omega}2)}
  • {1`2{1`2`}2} has recursion level {\theta(\Omega^\Omega,\omega)=\psi(\Omega^{\Omega^\Omega}\omega)}
  • {1`1`2{1`2`}2} has recursion level {\theta(\Omega^\Omega+1,0)=\psi(\Omega^{\Omega^\Omega+1})}
  • {1`1`1`2{1`2`}2} has recursion level {\theta(\Omega^\Omega+\Omega,0)=\psi(\Omega^{\Omega^\Omega+\Omega})}
  • {1{2`}2{1`2`}2} has recursion level {\theta(\Omega^\Omega+\Omega^\omega,0)=\psi(\Omega^{\Omega^\Omega+\Omega^\omega})}
  • {1{1{1{1`2`}2}2`}1{1`2`}2} has recursion level {\theta(\Omega^\Omega,1)=\psi(\Omega^{\Omega^\Omega}2)}
  • {1{1{1{1`2`}2}2`}2{1`2`}2} has recursion level {\theta(\Omega^\Omega+\Omega^{\theta(\Omega^\Omega,0)},0)=\psi(\Omega^{\Omega^\Omega+\Omega^{\psi(\Omega^{\Omega^\Omega})}})}
  • {1{1`2`}3} has recursion level {\theta(\Omega^\Omega2,0)=\psi(\Omega^{\Omega^\Omega2})}
  • {1{1`2`}1`2} has recursion level {\theta(\Omega^{\Omega+1},0)=\psi(\Omega^{\Omega^{\Omega+1}})}
  • {1{1`2`}1{2`}2} has recursion level {\theta(\Omega^{\Omega+\omega},0)=\psi(\Omega^{\Omega^{\Omega+\omega}})}
  • {1{1`2`}1{1`2`}2} has recursion level {\theta(\Omega^{\Omega2},0)=\psi(\Omega^{\Omega^{\Omega2}})}
  • {1{2`2`}2} has recursion level {\theta(\Omega^{\Omega\omega},0)=\psi(\Omega^{\Omega^{\Omega\omega}})}
  • {1{1`3`}2} has recursion level {\theta(\Omega^{\Omega^2},0)=\psi(\Omega^{\Omega^{\Omega^2}})}
  • {1{1`1,2`}2} has recursion level {\theta(\Omega^{\Omega^\omega},0)=\psi(\Omega^{\Omega^{\Omega^\omega}})}
  • {1{1`1`2`}2} has recursion level {\theta(\Omega^{\Omega^\Omega},0)=\psi(\Omega^{\Omega^{\Omega^\Omega}})}
  • {1{1`1`1`2`}2} has recursion level {\theta(\Omega^{\Omega^{\Omega^2}},0)=\psi(\Omega^{\Omega^{\Omega^{\Omega^2}}})}
  • {1{1{2`}2`}2} has recursion level {\theta(\Omega^{\Omega^{\Omega^\omega}},0)=\psi(\Omega^{\Omega^{\Omega^{\Omega^\omega}}})}
  • {1{1{1`2`}2`}2} has recursion level {\theta(\Omega^{\Omega^{\Omega^\Omega}},0)=\psi(\Omega^{\Omega^{\Omega^{\Omega^\Omega}}})}
  • {1{1{1`1`2`}2`}2} has recursion level {\theta(\Omega^{\Omega^{\Omega^{\Omega^\Omega}}},0)=\psi(\Omega^{\Omega^{\Omega^{\Omega^{\Omega^\Omega}}}})}
  • {1{1{1{1`2`}2`}2`}2} has recursion level {\theta(\Omega^{\Omega^{\Omega^{\Omega^{\Omega^\Omega}}}},0)=\psi(\Omega^{\Omega^{\Omega^{\Omega^{\Omega^{\Omega^\Omega}}}}})}

We can see that the strength of 1-separator was previously underestimated by 1 level of exponent tower inside ordinal collapsing functions such as ψ or θ.

By the way, array notation with this extra Rule still fits these comparisons.

  • Rule 4: {# A 1 B #′} = {# B #′} if lv(A) < lv(B)

Parts with the Rule 4 are named exAN′, EAN′, mEAN′, pDAN′, sDAN′ and DAN′ (corresponding with exAN ~ DAN). The Rule 4 doesn’t apply at the base layer, because the syntax must contain s(a,b #), and when b = 1, the rule at base layer will result invalid syntax.

sDAN vs. Taranovsky’s ordinal notation

Here’re the comparisons between sDAN and Taranovsky’s ordinal notation.

Fundamental sequences: Let L(α) be the amount of C’s in standard representation of α, then {\alpha[n]=\max\{\beta|\beta<\alpha\land L(\beta)\le L(\alpha)+n\}}.

Up to {1,,`1,,`2,,}

In last post, I showed that C(Ω22,0) works as the 1-separator in C(Ω2+____,β), but it’s C(Ω2+C(Ω2,C(Ω22,0)),0) that works as the 1-separator in C(Ω2+C(C(Ω22+____,0),C(Ω22,0)),β). And, C(Ω2,C(Ω22,0)) works as the 2-separator in C(Ω2+____,β).

Up to a C(Ω2,C(Ω2,C(Ω22,0)))

Now let ordinal σ = C(Ω22,0). And α+ is shorthand for C(Ω2,α).

  • {1{1`,,2,,}2} has recursion level C(C(Ω22+C(Ω2+C(σ+++),0),0),0) (`,, is the 1-separator on the double comma)
  • {1{1{1`,,2,,}2}2{1`,,2,,}2} has recursion level C(C(Ω22+C(Ω2+C(σ+++),0)2,0),0)
  • {1,,2{1`,,2,,}2} has recursion level C(C(Ω22+C(C(Ω2+C(σ+++),0)+,C(Ω2+C(σ+++),0)),0),0)
  • {1,,1,,2{1`,,2,,}2} has recursion level C(C(Ω22+C(σ,C(Ω2+C(σ+++),0)),0),0)
  • {1,,1,,1,,2{1`,,2,,}2} has recursion level C(C(Ω22+C(Ω2++σ,C(Ω2+C(σ+++),0)),0),0)
  • {1{1,,2,,}2{1`,,2,,}2} has recursion level C(C(Ω22+C(Ω2+ω^(σ++σ),C(Ω2+C(σ+++),0)),0),0)
  • {1{1,,1,,2,,}2{1`,,2,,}2} has recursion level C(C(Ω22+C(Ω2+ω^ω^(σ++σ),C(Ω2+C(σ+++),0)),0),0)
  • {1{1{1,,2,,}2,,}2{1`,,2,,}2} has recursion level C(C(Ω22+C(Ω2+ω^ω^ω^(σ++σ),C(Ω2+C(σ+++),0)),0),0)
  • {1{1`,,2,,}3} has recursion level C(C(Ω22+C(Ω2+C(σ+++),C(Ω2+C(σ+++),0)),0),0)
  • {1{1`,,2,,}1,2} has recursion level C(C(Ω22+C(Ω2+C(σ+++)+1,0),0),0)
  • {1{1`,,2,,}1,,2} has recursion level C(C(Ω22+C(Ω2+C(σ+++)+σ,0),0),0)
  • {1{1`,,2,,}1,,1,,2} has recursion level C(C(Ω22+C(Ω2+C(σ+++)+σ++σ,0),0),0)
  • {1{1`,,2,,}1{1`,,2,,}2} has recursion level C(C(Ω22+C(Ω2+C(σ+++)2,0),0),0)
  • {1{2`,,2,,}2} has recursion level C(C(Ω22+C(Ω2+ω^(C(σ+++)+1),0),0),0)
  • {1,,2`,,2,,} has recursion level C(C(Ω22+C(Ω2+ω^(C(σ+++)+σ),0),0),0)
  • {1,,1,,2`,,2,,} has recursion level C(C(Ω22+C(Ω2+ω^(C(σ+++)+σ++σ),0),0),0)
  • {1{1`,,2,,}2`,,2,,} has recursion level C(C(Ω22+C(Ω2+ω^(C(σ+++)2),0),0),0)
  • {1{1`,,2,,}1{1`,,2,,}2`,,2,,} has recursion level C(C(Ω22+C(Ω2+ω^ω^(C(σ+++)2),0),0),0)
  • {1{1{1`,,2,,}2`,,2,,}2`,,2,,} has recursion level C(C(Ω22+C(Ω2+ω^ω^ω^(C(σ+++)2),0),0),0)
  • {1`,,3,,} has recursion level C(C(Ω22+C(Ω2+C(σ++,C(σ+++)),0),0),0)
  • {1`,,1,2,,} has recursion level C(C(Ω22+C(Ω2+C(σ+++1,σ+),0),0),0)
  • {1`,,1{1`,,2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(σ+++C(σ+++),σ+),0),0),0)
  • {1`,,1`,,2,,} has recursion level C(C(Ω22+C(Ω2+C(σ++2,σ+),0),0),0)
  • {1{2`,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(ω^(σ+++1),σ+),0),0),0)
  • {1{1`,,2`,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(ω^(σ++2),σ+),0),0),0)
  • {1{1{1`,,2`,,}2`,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(ω^ω^ω^(σ++2),σ+),0),0),0)
  • {1{1` `,,2`,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(σ++++),0),0),0) (` `,, is the 1-separator on the `,,)
  • {1,,`1,2,,} has recursion level C(C(Ω22+C(Ω2+C(C(Ω2+1,σ),σ+),0),0),0) (,,` is the 2-separator on the double comma)
  • {1,,`1,,2,,} has recursion level C(C(Ω22+C(Ω2+C(C(Ω2+σ,σ),σ+),0),0),0)
  • {1{1,,`1{1{1,,`1,,2,,}2}2,,}1,2} has recursion level C(C(Ω22+C(Ω2+C(C(Ω2+σ,σ),σ+)+1,0),0),0)
  • {1{1,,`1{1{1,,`1,,2,,}2}2,,}1{1,,`1,2,,}2} has recursion level C(C(Ω22+C(Ω2+C(C(Ω2+σ,σ),σ+)+C(C(Ω2+1,σ),σ+),0),0),0)
  • {1{1,,`1{1{1,,`1,,2,,}2}2,,}1{1,,`1{1{1,,`1,,2,,}2}2,,}2} has recursion level C(C(Ω22+C(Ω2+C(C(Ω2+σ,σ),σ+)2,0),0),0)
  • {1{2{1,,`1{1{1,,`1,,2,,}2}2,,}2,,}2} has recursion level C(C(Ω22+C(Ω2+ω^(C(C(Ω2+σ,σ),σ+)+1),0),0),0)
  • {1`,,2{1,,`1{1{1,,`1,,2,,}2}2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(σ++,C(C(Ω2+σ,σ),σ+)),0),0),0)
  • {1{1,,`1{1{1,,`1,,2,,}2}2,,}1,2,,} has recursion level C(C(Ω22+C(Ω2+C(C(Ω2+σ,σ)+1,σ+),0),0),0)
  • {1{1,,`1,2,,}2,,`1{1{1,,`1,,2,,}2}2,,} has recursion level C(C(Ω22+C(Ω2+C(C(Ω2+σ,σ)+C(Ω2+1,σ),σ+),0),0),0)
  • {1{1,,`1{1{1,,`1,,2,,}2}2,,}2,,`1{1{1,,`1,,2,,}2}2,,} has recursion level C(C(Ω22+C(Ω2+C(C(Ω2+σ,σ)2,σ+),0),0),0)
  • {1,,`2{1{1,,`1,,2,,}2}2,,} has recursion level C(C(Ω22+C(Ω2+C(C(Ω2+σ,σ)++),0),0),0)
  • {1,,`1{1{1,,`1,,2,,}2}1,2,,} has recursion level C(C(Ω22+C(Ω2+C(C(Ω2+σ+1,σ),σ+),0),0),0)
  • {1,,`1{1{1,,`1,,2,,}2}1{1{1,,`1,,2,,}2}2,,} has recursion level C(C(Ω22+C(Ω2+C(C(Ω2+σ2,σ),σ+),0),0),0)
  • {1,,2{1,,`1,,2,,}2} has recursion level C(C(Ω22+C(Ω2+C(C(Ω2+C(σ+,σ),σ),σ+),0),0),0)
  • {1,,1,2{1,,`1,,2,,}2} has recursion level C(C(Ω22+C(Ω2+C(C(Ω2+C(C(Ω2+1,σ),σ),σ),σ+),0),0),0)
  • {1,,1{1{1,,`1,,2,,}2}2{1,,`1,,2,,}2} has recursion level C(C(Ω22+C(Ω2+C(C(Ω2+C(C(Ω2+σ,σ),σ),σ),σ+),0),0),0)
  • {1,,1{1,,2{1,,`1,,2,,}2}2{1,,`1,,2,,}2} has recursion level C(C(Ω22+C(Ω2+C(C(Ω2+C(C(Ω2+,σ),σ),σ),σ+),0),0),0)
  • {1,,1{1{1,,1,2{1,,`1,,2,,}2}2,,2{1,,`1,,2,,}2}2{1,,`1,,2,,}2} has recursion level C(C(Ω22+C(Ω2+C(C(Ω2+C(C(Ω2+C(C(Ω2+1,σ),σ+),σ),σ),σ),σ+),0),0),0)
  • {1,,1{1{1,,1{1{1,,`1,,2,,}2}2{1,,`1,,2,,}2}2,,2{1,,`1,,2,,}2}2{1,,`1,,2,,}2} has recursion level C(C(Ω22+C(Ω2+C(C(Ω2+C(C(Ω2+C(C(Ω2+σ,σ),σ+),σ),σ),σ),σ+),0),0),0)
  • {1,,1{1{1,,1{1,,2{1,,`1,,2,,}2}2{1,,`1,,2,,}2}2,,2{1,,`1,,2,,}2}2{1,,`1,,2,,}2} has recursion level C(C(Ω22+C(Ω2+C(C(Ω2+C(C(Ω22,σ),σ),σ),σ+),0),0),0)
  • {1,,2{1{1,,1{1,,2{1,,`1,,2,,}2}2{1,,`1,,2,,}2}2,,2{1,,`1,,2,,}2}2{1,,`1,,2,,}2} has recursion level C(C(Ω22+C(Ω2+C(C(Ω2+C(C(Ω22,σ)+,σ),σ),σ+),0),0),0)
  • {1,,1{1{1,,1{1,,2{1,,`1,,2,,}2}2{1,,`1,,2,,}2}2,,2{1,,`1,,2,,}2}1,2{1,,`1,,2,,}2} has recursion level C(C(Ω22+C(Ω2+C(C(Ω2+C(C(Ω22+1,0),σ),σ),σ+),0),0),0)
  • {1,,1{1{1,,1{1,,2{1,,`1,,2,,}2}2{1,,`1,,2,,}2}2,,2{1,,`1,,2,,}2}1{1{1,,`1{1{1,,`1,,2,,}2}2,,}2}2{1,,`1,,2,,}2} has recursion level C(C(Ω22+C(Ω2+C(C(Ω2+C(C(Ω22+C(Ω2+C(C(Ω2+σ,σ),σ+),0),0),σ),σ),σ+),0),0),0)
  • {1,,1{1{1,,1{1,,2{1,,`1,,2,,}2}2{1,,`1,,2,,}2}2,,2{1,,`1,,2,,}2}1{1{1,,`1,,2,,}2}2{1,,`1,,2,,}2} has recursion level C(C(Ω22+C(Ω2+C(C(Ω2+,σ),σ+),0),0),0)
  • {1{1,,`1,,2,,}1,2} has recursion level C(C(Ω22+C(Ω2+C(C(Ω2+,σ),σ+)+1,0),0),0)
  • {1`,,2{1,,`1,,2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(σ++,C(C(Ω2+,σ),σ+)),0),0),0)
  • {1{1,,`1,,2,,}1,2,,} has recursion level C(C(Ω22+C(Ω2+C(C(Ω2+,σ)+1,σ+),0),0),0)
  • {1{1,,`1,2,,}2,,`1,,2,,} has recursion level C(C(Ω22+C(Ω2+C(C(Ω2+,σ)+C(Ω2+1,σ),σ+),0),0),0)
  • {1,,`2,,2,,} has recursion level C(C(Ω22+C(Ω2+C(C(Ω2+,σ)++),0),0),0)
  • {1,,`1,,1,2,,} has recursion level C(C(Ω22+C(Ω2+C(C(Ω2++1,σ),σ+),0),0),0)
  • {1,,`1{1`,,2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(C(Ω2+C(σ+++),σ),σ+),0),0),0)
  • {1,,`1{1{1,,`1,2,,}2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(C(Ω2+C(C(Ω2+1,σ),σ+),σ),σ+),0),0),0)
  • {1,,`1`,,2,,} has recursion level C(C(Ω22+C(Ω2+C(C(Ω22,σ),σ+),0),0),0)
  • {1{1,,`1`,,2,,}1,2} has recursion level C(C(Ω22+C(Ω2+C(C(Ω22,σ),σ+)+1,0),0),0)
  • {1`,,2{1,,`1`,,2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(σ++,C(C(Ω22,σ),σ+)),0),0),0)
  • {1{1,,`1`,,2,,}3,,} has recursion level C(C(Ω22+C(Ω2+C(C(Ω22,σ),C(C(Ω22,σ),σ+)),0),0),0)
  • {1{1,,`1`,,2,,}1,2,,} has recursion level C(C(Ω22+C(Ω2+C(C(Ω22,σ)+1,σ+),0),0),0)
  • {1{1,,`1`,,2,,}2,,`1`,,2,,} has recursion level C(C(Ω22+C(Ω2+C(C(Ω22,σ)2,σ+),0),0),0)
  • {1,,`2`,,2,,} has recursion level C(C(Ω22+C(Ω2+C(C(Ω22,σ)++),0),0),0)
  • {1,,`1{1,,`2`,,2,,}1`,,2,,} has recursion level C(C(Ω22+C(Ω2+C(C(Ω2+C(C(Ω22,σ)++),C(Ω22,σ)),σ+),0),0),0)
  • {1,,`1`,,3,,} has recursion level C(C(Ω22+C(Ω2+C(C(Ω22,C(Ω22,σ)),σ+),0),0),0)
  • {1,,`1`,,1,2,,} has recursion level C(C(Ω22+C(Ω2+C(C(Ω22+1,0),σ+),0),0),0)
  • {1,,`1`,,1{1{2,,}2}2,,} has recursion level C(C(Ω22+C(Ω2+C(C(Ω22+C(Ω2+ω^(σ++1),0),0),σ+),0),0),0)
  • {1,,`1`,,1{1{1,,`1`,,1,2,,}2}2,,} has recursion level C(C(Ω22+C(Ω2+C(C(Ω22+C(Ω2+C(C(Ω22+1,0),σ+),0),0),σ+),0),0),0)
  • {1,,`1`,,1,,2,,} has recursion level C(C(Ω22+C(Ω2++,0),0),0)

So here comes a further guess: does C(Ω2+C(Ω2,C(Ω2,C(Ω22,0))),0) work as the 2-separator in C(Ω2+C(C(Ω22+____,0),C(Ω22,0)),β)?

Up to a C(Ω2,C(Ω2,C(Ω2,C(Ω22,0))))

Now let separator ♦ = {1{1,,`1`,,1,,2,,}2} and let ordinal σ = C(Ω22,0). And α+ is shorthand for C(Ω2,α).

  • {1,,1{1{1{1,,`1`,,1♦2,,}2}2,,1{1,,1,,2}2}2{1,,1,,2}2} has recursion level C(C(Ω22+C(Ω2++,0),0),0)
  • {1,,1{1{1{1,,`1`,,1♦2,,}2}2,,1{1,,1,,2}2}1{1{1{1,,`1`,,1♦2,,}2}2,,1{1,,1,,2}2}2{1,,1,,2}2} has recursion level C(C(Ω22+C(Ω2++,0)+C(C(Ω22+C(Ω2++,0),0),C(Ω2+σ,0)),0),0)
  • {1,,1{1{1{1,,`1`,,1♦2,,}2}1,2,,1{1,,1,,2}2}2{1,,1,,2}2} has recursion level C(C(Ω22+C(Ω2++,0)+C(C(Ω22+C(Ω2++,0),0)+1,C(Ω2+σ,0)),0),0)
  • {1{1,,1{1{1,,`1`,,1♦2,,}2}2}2,,1{1{1,,`1`,,1♦2,,}2}2} has recursion level C(C(Ω22+C(Ω2++,0)+C(C(Ω22+C(Ω2++,0),0)2,C(Ω2+σ,0)),0),0)
  • {1,,2{1{1,,`1`,,1♦2,,}2}2} has recursion level C(C(Ω22+C(Ω2++,0)+C(C(Ω22+C(Ω2++,0),0)+,C(Ω2+σ,0)),0),0)
  • {1,,1{1,,1{1{1,,`1`,,1♦2,,}2}2}1,2{1{1,,`1`,,1♦2,,}2}2} has recursion level C(C(Ω22+C(Ω2++,0)+C(C(Ω22+C(Ω2++,0)+1,0),C(Ω2+σ,0)),0),0)
  • {1,,1{1,,1{1{1,,`1`,,1♦2,,}2}2}1{1,,2{1,,1,,2}2}2{1{1,,`1`,,1♦2,,}2}2} has recursion level C(C(Ω22+C(Ω2++,0)+C(Ω2+σ,0)+,0),0)
  • {1,,1{1,,1{1{1,,`1`,,1♦2,,}2}2}1{1,,1{1{1,,`1`,,1♦2,,}2}2}2{1{1,,`1`,,1♦2,,}2}2} has recursion level C(C(Ω22+C(Ω2++,0)+C(Ω2+C(C(Ω22+C(Ω2++,0),0),σ),0),0),0)
  • {1,,1{2{1,,`1`,,1♦2,,}2}2} has recursion level C(C(Ω22+C(Ω2++,0)+C(Ω2+ω^(C(C(Ω22+C(Ω2++,0),0),σ)+1),0),0),0)
  • {1{1,,1,,2}2{1,,`1`,,1♦2,,}2} has recursion level C(C(Ω22+C(Ω2++,0)+C(Ω2+ω^(C(C(Ω22+C(Ω2++,0),0),σ)+σ),0),0),0)
  • {1{1,,1,,1,2}2{1,,`1`,,1♦2,,}2} has recursion level C(C(Ω22+C(Ω2++,0)+C(Ω2++1,0),0),0)
  • {1{1,,1,,1,,1,2}2{1,,`1`,,1♦2,,}2} has recursion level C(C(Ω22+C(Ω2++,0)+C(Ω2+2+1,0),0),0)
  • {1{1{1`,,2,,}2}2{1,,`1`,,1♦2,,}2} has recursion level C(C(Ω22+C(Ω2++,0)+C(Ω2+C(σ+++),0),0),0)
  • {1{1{1,,`1,2,,}2}2{1,,`1`,,1♦2,,}2} has recursion level C(C(Ω22+C(Ω2++,0)+C(Ω2+C(C(Ω2+1,σ),σ+),0),0),0)
  • {1{1{1,,`1`,,2,,}2}2{1,,`1`,,1♦2,,}2} has recursion level C(C(Ω22+C(Ω2++,0)+C(Ω2+C(C(Ω22,σ),σ+),0),0),0)
  • {1{1{1,,`1`,,1,2,,}2}2{1,,`1`,,1♦2,,}2} has recursion level C(C(Ω22+C(Ω2++,0)+C(Ω2+C(C(Ω22+1,0),σ+),0),0),0)
  • {1{1{1,,`1`,,1♦2,,}2}2{1,,`1`,,1♦2,,}2} has recursion level C(C(Ω22+C(Ω2++,0)+C(Ω2+C(C(Ω22+C(Ω2++,0),0),σ+),0),0),0)
  • {1{1{1,,`1`,,1♦2,,}2}1,2{1,,`1`,,1♦2,,}2} has recursion level C(C(Ω22+C(Ω2++,0)+ω^(C(Ω2+C(C(Ω22+C(Ω2++,0),0),σ+),0)+1),0),0)
  • {1,,2{1,,`1`,,1♦2,,}2} has recursion level C(C(Ω22+C(Ω2++,0)+C(C(Ω2+C(C(Ω22+C(Ω2++,0),0),σ+),0)+,C(Ω2+C(C(Ω22+C(Ω2++,0),0),σ+),0)),0),0)
  • {1,,1,,2{1,,`1`,,1♦2,,}2} has recursion level C(C(Ω22+C(Ω2++,0)+C(σ,C(Ω2+C(C(Ω22+C(Ω2++,0),0),σ+),0)),0),0)
  • {1,,1{1,,2,,2{1,,`1`,,1♦2,,}2}1{1,,1,,2{1,,`1`,,1♦2,,}2}2,,2{1,,`1`,,1♦2,,}2} has recursion level C(C(Ω22+C(Ω2++,0)+C(Ω2+C(Ω22+C(Ω2+,C(Ω2+C(C(Ω22+C(Ω2++,0),0),σ+),0)),σ),C(Ω2+C(C(Ω22+C(Ω2++,0),0),σ+),0)),0),0)
  • {1,,1,,1,2{1,,`1`,,1♦2,,}2} has recursion level C(C(Ω22+C(Ω2++,0)+C(Ω2+C(Ω22+C(Ω2++1,C(Ω2+C(C(Ω22+C(Ω2++,0),0),σ+),0)),σ),C(Ω2+C(C(Ω22+C(Ω2++,0),0),σ+),0)),0),0)
  • {1,,1,,1,,2{1,,`1`,,1♦2,,}2} has recursion level C(C(Ω22+C(Ω2++,0)+C(Ω2+C(Ω22+C(Ω2++σ,C(Ω2+C(C(Ω22+C(Ω2++,0),0),σ+),0)),σ),C(Ω2+C(C(Ω22+C(Ω2++,0),0),σ+),0)),0),0)
  • {1{1,,`1`,,1,2,,}2{1,,`1`,,1♦2,,}2} has recursion level C(C(Ω22+C(Ω2++,0)+C(Ω2+C(Ω22+C(Ω2+C(C(Ω22+1,0),σ+),C(Ω2+C(C(Ω22+C(Ω2++,0),0),σ+),0)),σ),C(Ω2+C(C(Ω22+C(Ω2++,0),0),σ+),0)),0),0)
  • {1{1,,`1`,,1♦2,,}3} has recursion level C(C(Ω22+C(Ω2++,0)+C(Ω2+C(C(Ω22+C(Ω2++,0),0),σ),C(Ω2+C(C(Ω22+C(Ω2++,0),0),σ+),0)),0),0)
  • {1{1{1,,`1`,,1♦2,,}3}2{1,,`1`,,1♦2,,}3} has recursion level C(C(Ω22+C(Ω2++,0)+C(Ω2+C(C(Ω22+C(Ω2++,0),0),σ+),C(Ω2+C(C(Ω22+C(Ω2++,0),0),σ+),0)),0),0)
  • {1{1,,`1`,,1♦2,,}1,2} has recursion level C(C(Ω22+C(Ω2++,0)+C(Ω2+C(C(Ω22+C(Ω2++,0),0),σ+)+1,0),0),0)
  • {1{1,,`1`,,1♦2,,}1,,2} has recursion level C(C(Ω22+C(Ω2++,0)+C(Ω2+C(C(Ω22+C(Ω2++,0),0),σ+)+σ,0),0),0)
  • {1{1,,`1`,,1♦2,,}1,,1,2} has recursion level C(C(Ω22+C(Ω2++,0)+C(Ω2+C(C(Ω22+C(Ω2++,0),0),σ+)+σ++1,0),0),0)
  • {1{1,,`1`,,1♦2,,}1{1,,`1`,,1,2,,}2} has recursion level C(C(Ω22+C(Ω2++,0)+C(Ω2+C(C(Ω22+C(Ω2++,0),0),σ+)+C(C(Ω22+1,0),σ+),0),0),0)
  • {1{1,,`1`,,1♦2,,}1{1,,`1`,,1{1{1,,`1`,,1♦2,,}2}2,,}2} has recursion level C(C(Ω22+C(Ω2++,0)+C(Ω2+C(C(Ω22+C(Ω2++,0),0),σ+)+C(C(Ω22+C(Ω2+C(C(Ω22+C(Ω2++,0),0),σ+),0),0),σ+),0),0),0)
  • {1{1,,`1`,,1♦2,,}1{1,,`1`,,1♦2,,}2} has recursion level C(C(Ω22+C(Ω2++,0)+C(Ω2+C(C(Ω22+C(Ω2++,0),0),σ+)2,0),0),0)
  • {1{2{1,,`1`,,1♦2,,}2,,}2} has recursion level C(C(Ω22+C(Ω2++,0)+C(Ω2+ω^(C(C(Ω22+C(Ω2++,0),0),σ+)+1),0),0),0)
  • {1`,,2{1,,`1`,,1♦2,,}2,,} has recursion level C(C(Ω22+C(Ω2++,0)+C(Ω2+C(σ++,C(C(Ω22+C(Ω2++,0),0),σ+)),0),0),0)
  • {1{1,,`1,,1,2,,}2{1,,`1`,,1♦2,,}2,,} has recursion level C(C(Ω22+C(Ω2++,0)+C(Ω2+C(C(Ω2++1,σ),C(C(Ω22+C(Ω2++,0),0),σ+)),0),0),0)
  • {1{1,,`1`,,2,,}2{1,,`1`,,1♦2,,}2,,} has recursion level C(C(Ω22+C(Ω2++,0)+C(Ω2+C(C(Ω22,σ),C(C(Ω22+C(Ω2++,0),0),σ+)),0),0),0)
  • {1{1,,`1`,,1♦2,,}3,,} has recursion level C(C(Ω22+C(Ω2++,0)+C(Ω2+C(C(Ω22+C(Ω2++,0),0),C(C(Ω22+C(Ω2++,0),0),σ+)),0),0),0)
  • {1{1,,`1`,,1♦2,,}1,2,,} has recursion level C(C(Ω22+C(Ω2++,0)+C(Ω2+C(C(Ω22+C(Ω2++,0),0)+1,σ+),0),0),0)
  • {1{1,,`1,2,,}2,,`1`,,1♦2,,} has recursion level C(C(Ω22+C(Ω2++,0)+C(Ω2+C(C(Ω22+C(Ω2++,0),0)+C(Ω2+1,σ),σ+),0),0),0)
  • {1{1,,`1`,,1♦2,,}2,,`1`,,1♦2,,} has recursion level C(C(Ω22+C(Ω2++,0)+C(Ω2+C(C(Ω22+C(Ω2++,0),0)2,σ+),0),0),0)
  • {1,,`2`,,1♦2,,} has recursion level C(C(Ω22+C(Ω2++,0)+C(Ω2+C(C(Ω22+C(Ω2++,0),0)++),0),0),0)
  • {1,,`1`,,2♦2,,} has recursion level C(C(Ω22+C(Ω2++,0)+C(Ω2+C(C(Ω22,C(Ω22+C(Ω2++,0),0)),σ+),0),0),0)
  • {1,,`1`,,1♦1,2,,} has recursion level C(C(Ω22+C(Ω2++,0)+C(Ω2+C(C(Ω22+C(Ω2++,0)+1,0),σ+),0),0),0)
  • {1,,`1`,,1♦1{1{1,,`1`,,1♦2,,}2}2,,} has recursion level C(C(Ω22+C(Ω2++,0)+C(Ω2+C(C(Ω22+C(Ω2++,0)+C(Ω2+C(C(Ω22+C(Ω2++,0),0),σ+),0),0),σ+),0),0),0)
  • {1,,`1`,,1♦1♦2,,} has recursion level C(C(Ω22+C(Ω2++,0)2,0),0)
  • {1,,2{1,,`1`,,1,,2,,}2} has recursion level C(C(Ω22+C(C(Ω2++,0)+,C(Ω2++,0)),0),0)
  • {1,,1,,2{1,,`1`,,1,,2,,}2} has recursion level C(C(Ω22+C(σ,C(Ω2++,0)),0),0)
  • {1,,1,,1,,2{1,,`1`,,1,,2,,}2} has recursion level C(C(Ω22+C(Ω2++σ,C(Ω2++,0)),0),0)
  • {1{1`,,2,,}2{1,,`1`,,1,,2,,}2} has recursion level C(C(Ω22+C(Ω2+C(σ+++),C(Ω2++,0)),0),0)
  • {1{1,,`1,2,,}2{1,,`1`,,1,,2,,}2} has recursion level C(C(Ω22+C(Ω2+C(C(Ω2+1,σ),σ+),C(Ω2++,0)),0),0)
  • {1{1,,`1,,1,2,,}2{1,,`1`,,1,,2,,}2} has recursion level C(C(Ω22+C(Ω2+C(C(Ω2++1,σ),σ+),C(Ω2++,0)),0),0)
  • {1{1,,`1{1{1,,`1,2,,}2,,}2,,}2{1,,`1`,,1,,2,,}2} has recursion level C(C(Ω22+C(Ω2+C(C(Ω2+C(C(Ω2+1,σ),σ+),σ),σ+),C(Ω2++,0)),0),0)
  • {1{1,,`1`,,2,,}2{1,,`1`,,1,,2,,}2} has recursion level C(C(Ω22+C(Ω2+C(C(Ω22,σ),σ+),C(Ω2++,0)),0),0)
  • {1{1,,`1`,,1,2,,}2{1,,`1`,,1,,2,,}2} has recursion level C(C(Ω22+C(Ω2+C(C(Ω22+1,0),σ+),C(Ω2++,0)),0),0)
  • {1{1,,`1`,,1{1{1,,`1`,,1♦2,,}2}2,,}2{1,,`1`,,1,,2,,}2} has recursion level C(C(Ω22+C(Ω2+C(C(Ω22+C(Ω2+C(C(Ω22+C(Ω2++,0),0),σ+),0),0),σ+),C(Ω2++,0)),0),0)
  • {1{1,,`1`,,1♦2,,}2{1,,`1`,,1,,2,,}2} has recursion level C(C(Ω22+C(Ω2+C(C(Ω22+C(Ω2++,0),0),σ+),C(Ω2++,0)),0),0)
  • {1{1,,`1`,,1{1,,2{1,,`1`,,1,,2,,}2}2,,}2{1,,`1`,,1,,2,,}2} has recursion level C(C(Ω22+C(Ω2+C(C(Ω22+C(Ω2++,0)+,0),σ+),C(Ω2++,0)),0),0)
  • {1{1,,`1`,,1,,2,,}3} has recursion level C(C(Ω22+C(Ω2++,C(Ω2++,0)),0),0)
  • {1{1,,`1`,,1,,2,,}1,2} has recursion level C(C(Ω22+C(Ω2+++1,0),0),0)

As shown above, C(Ω2+C(Ω2,C(Ω2,C(Ω22,0))),0) is not so strong as I originally conjectured – it just works as the 1-separator in C(Ω2+C(C(Ω22+____,0),C(Ω2,C(Ω22,0))),β), and corresponds to {1{1,,`1`,,1,,2,,}2}.

Now let ordinal σ = C(Ω22,0). And α+ is shorthand for C(Ω2,α).

  • {1{1,,`1`,,1,,2,,}1,,2} has recursion level C(C(Ω22+C(Ω2+++σ,0),0),0)
  • {1{1,,`1`,,1,,2,,}1{1{1,,`1`,,1,,2,,}1,,2}1{1{1,,`1`,,1,,2,,}1,,2}2} has recursion level C(C(Ω22+C(Ω2+++σ2,0),0),0)
  • {1,,2{1,,`1`,,1,,2,,}1,,2} has recursion level C(C(Ω22+C(Ω2+++C(σ+,σ),0),0),0)
  • {1,,1{1{1,,`1`,,1,,2,,}1,,2}2{1,,`1`,,1,,2,,}1,,2} has recursion level C(C(Ω22+C(Ω2+++C(C(Ω2+σ,σ),σ),0),0),0)
  • {1,,1{1,,2{1,,`1`,,1,,2,,}1,,2}2{1,,`1`,,1,,2,,}1,,2} has recursion level C(C(Ω22+C(Ω2+++C(C(Ω2+,σ),σ),0),0),0)
  • {1,,1{1,,3{1,,`1`,,1,,2,,}1,,2}2{1,,`1`,,1,,2,,}1,,2} has recursion level C(C(Ω22+C(Ω2+++C(C(Ω2++,σ),σ),0),0),0)
  • {1,,1{1,,3{1,,`1`,,1,,2,,}1,,2}1{1,,2{1,,`1`,,1,,2,,}1,,2}2{1,,`1`,,1,,2,,}1,,2} has recursion level C(C(Ω22+C(Ω2+++C(C(Ω22,σ),σ),0),0),0)
  • {1,,1{1,,3{1,,`1`,,1,,2,,}1,,2}1{1,,2{1,,`1`,,1,,2,,}1,,2}1,2{1,,`1`,,1,,2,,}1,,2} has recursion level C(C(Ω22+C(Ω2+++C(C(Ω22+1,0),σ),0),0),0)
  • {1,,1{1,,3{1,,`1`,,1,,2,,}1,,2}1{1,,2{1,,`1`,,1,,2,,}1,,2}1{1{1,,`1`,,1,,2,,}1,,2}2{1,,`1`,,1,,2,,}1,,2} has recursion level C(C(Ω22+C(Ω2+++,0),0),0)
  • {1{1,,`1`,,1,,2,,}1,,1,2} has recursion level C(C(Ω22+C(Ω2++++1,0),0),0)
  • {1{1,,`1`,,1,,2,,}1{1,,`1,,2,,}2} has recursion level C(C(Ω22+C(Ω2+++C(C(Ω2+σ,σ),σ+),0),0),0)
  • {1{1,,`1`,,1,,2,,}1{1,,`1,,1,2,,}2} has recursion level C(C(Ω22+C(Ω2+++C(C(Ω2++1,σ),σ+),0),0),0)
  • {1{1,,`1`,,1,,2,,}1{1,,`1`,,2,,}2} has recursion level C(C(Ω22+C(Ω2+++C(C(Ω2++,σ),σ+),0),0),0)
  • {1{1,,`1`,,1,,2,,}1{1,,`1`,,1,,2,,}2} has recursion level C(C(Ω22+C(Ω2+++C(C(Ω2+++σ,σ),σ+),0),0),0) (here Taranovsky’s ordinal notation has an unexpected growth boost because C(Ω2+++C(C(Ω2++,σ),σ+),0) is valid)
  • {1,,`1`,,1,,1,2,,} has recursion level C(C(Ω22+C(Ω2+++C(C(Ω2++++1,σ),σ+),0),0),0)
  • {1,,`1`,,1`,,2,,} has recursion level C(C(Ω22+C(Ω2+++C(C(Ω22,σ),σ+),0),0),0)
  • {1,,`1`,,1`,,1,2,,} has recursion level C(C(Ω22+C(Ω2+++C(C(Ω22+1,0),σ+),0),0),0)
  • {1,,`1`,,1`,,1,,2,,} has recursion level C(C(Ω22+C(Ω2++2,0),0),0)
  • {1{1,,`1`,,1`,,1,,2,,}1,2} has recursion level C(C(Ω22+C(Ω2++2+1,0),0),0)
  • {1{1,,`1`,,1`,,1,,2,,}1{1,,`1,,1,2,,}2} has recursion level C(C(Ω22+C(Ω2++2+C(C(Ω2++1,σ),σ+),0),0),0)
  • {1{1,,`1`,,1`,,1,,2,,}1{1,,`1`,,2,,}2} has recursion level C(C(Ω22+C(Ω2++2+C(C(Ω2++,σ),σ+),0),0),0)
  • {1{1,,`1`,,1`,,1,,2,,}1{1,,`1`,,1`,,2,,}2} has recursion level C(C(Ω22+C(Ω2++2+C(C(Ω2++2,σ),σ+),0),0),0)
  • {1{1,,`1`,,1`,,1,,2,,}1{1,,`1`,,1`,,1,,2,,}2} has recursion level C(C(Ω22+C(Ω2++2+C(C(Ω2++2+σ,σ),σ+),0),0),0)
  • {1,,`1`,,1`,,1`,,2,,} has recursion level C(C(Ω22+C(Ω2++2+C(C(Ω22,σ),σ+),0),0),0)
  • {1,,`1`,,1`,,1`,,1,,2,,} has recursion level C(C(Ω22+C(Ω2++3,0),0),0)
  • {1,,`1`,,1`,,1`,,1`,,1,,2,,} has recursion level C(C(Ω22+C(Ω2++4,0),0),0)
  • {1,,`1{2`,,}2,,} has recursion level C(C(Ω22+C(Ω2+ω^(σ+++1),0),0),0)
  • {1,,`1{1,,2`,,}2,,} has recursion level C(C(Ω22+C(Ω2+ω^(σ+++σ),0),0),0)
  • {1{1,,`1{1{1{1,,`1{1,,2`,,}2,,}2}2`,,}2,,}1,2} has recursion level C(C(Ω22+C(Ω2+ω^(σ+++σ)+1,0),0),0)
  • {1{1,,`1{1,,2`,,}2,,}1,2} has recursion level C(C(Ω22+C(Ω2+ω^(σ+++)+1,0),0),0)
  • {1{1,,`1{1,,2`,,}2,,}1,,1,2} has recursion level C(C(Ω22+C(Ω2+ω^(σ+++)+σ++1,0),0),0)
  • {1{1,,`1{1,,2`,,}2,,}1{1,,`1`,,2,,}2} has recursion level C(C(Ω22+C(Ω2+ω^(σ+++)+C(C(Ω2++,σ),σ+),0),0),0)
  • {1{1,,`1{1,,2`,,}2,,}1{1,,`1{1,,2`,,}2,,}2} has recursion level C(C(Ω22+C(Ω2+ω^(σ+++)+C(C(Ω2+ω^(σ+++σ),σ),σ+),0),0),0)
  • {1,,`1{1,,2`,,}1`,,2,,} has recursion level C(C(Ω22+C(Ω2+ω^(σ+++)+C(C(Ω22,σ),σ+),0),0),0)
  • {1,,`1{1,,2`,,}1`,,1,,2,,} has recursion level C(C(Ω22+C(Ω2+ω^(σ+++)+σ++,0),0),0)
  • {1,,`1{2,,2`,,}2,,} has recursion level C(C(Ω22+C(Ω2+ω^(σ++++1),0),0),0)
  • {1,,`1{1,,3`,,}2,,} has recursion level C(C(Ω22+C(Ω2+ω^(σ++++σ),0),0),0)
  • {1,,`1{1{1{1,,`1`,,2,,}2,,}2`,,}2,,} has recursion level C(C(Ω22+C(Ω2+ω^(σ+++C(C(Ω2++,σ),σ+)),0),0),0)
  • {1,,`1{1{1{1,,`1{2,,2`,,}2,,}2,,}2`,,}2,,} has recursion level C(C(Ω22+C(Ω2+ω^(σ+++C(C(Ω2+ω^(σ++++1),σ),σ+)),0),0),0)
  • {1,,`1{1`,,2`,,}2,,} has recursion level C(C(Ω22+C(Ω2+ω^(σ+++C(C(Ω22,σ),σ+)),0),0),0)
  • {1{1,,`1{1`,,2`,,}2,,}1,2} has recursion level C(C(Ω22+C(Ω2+ω^(σ+++C(C(Ω22,σ),σ+))+1,0),0),0)
  • {1{1,,`1{1`,,2`,,}2,,}1{1,,`1{2`,,}2,,}2} has recursion level C(C(Ω22+C(Ω2+ω^(σ+++C(C(Ω22,σ),σ+))+C(C(Ω2+ω^(σ+++1),σ),σ+),0),0),0)
  • {1{1,,`1{1`,,2`,,}2,,}1{1,,`1{1{1{1,,`1{2`,,}2,,}2,,}2`,,}2,,}2} has recursion level C(C(Ω22+C(Ω2+ω^(σ+++C(C(Ω22,σ),σ+))+C(C(Ω2+ω^(σ+++C(C(Ω2+ω^(σ+++1),σ),σ+)),σ),σ+),0),0),0)
  • {1{1,,`1{1`,,2`,,}2,,}1{1,,`1{1`,,2`,,}2,,}2} has recursion level C(C(Ω22+C(Ω2+ω^(σ+++C(C(Ω22,σ),σ+))+C(C(Ω22,σ),σ+),0),0),0) (here an erratic thing comes because C(C(Ω2+ω^(σ+++C(C(Ω22,σ),σ+)),σ),σ+) is not valid)
  • {1{2{1,,`1{1`,,2`,,}2,,}2,,}2} has recursion level C(C(Ω22+C(Ω2+ω^(σ+++C(C(Ω22,σ),σ+))+ω^(C(C(Ω22,σ),σ+)+1),0),0),0)
  • {1`,,2{1,,`1{1`,,2`,,}2,,}2,,} has recursion level C(C(Ω22+C(Ω2+ω^(σ+++C(C(Ω22,σ),σ+))+C(σ++,C(C(Ω22,σ),σ+)),0),0),0)
  • {1{1,,`1{2`,,}2,,}2{1,,`1{1`,,2`,,}2,,}2,,} has recursion level C(C(Ω22+C(Ω2+ω^(σ+++C(C(Ω22,σ),σ+))+C(C(Ω2+ω^(σ+++1),σ),C(C(Ω22,σ),σ+)),0),0),0)
  • {1{1,,`1{1{1{1,,`1{2`,,}2,,}2,,}2`,,}2,,}2{1,,`1{1`,,2`,,}2,,}2,,} has recursion level C(C(Ω22+C(Ω2+ω^(σ+++C(C(Ω22,σ),σ+))+C(C(Ω2+ω^(σ+++C(C(Ω2+ω^(σ+++1),σ),σ+)),σ),C(C(Ω22,σ),σ+)),0),0),0)
  • {1{1,,`1{1{1{1,,`1{1`,,2`,,}2,,}2,,}2`,,}2,,}2{1,,`1{1`,,2`,,}2,,}2,,} has recursion level C(C(Ω22+C(Ω2+ω^(σ+++C(C(Ω22,σ),σ+))+C(C(Ω2+ω^(σ+++C(C(Ω22,σ),σ+)),σ),C(C(Ω22,σ),σ+)),0),0),0)
  • {1{1,,`1{1{1{1,,`1{1`,,2`,,}2,,}2,,}2`,,}1,2,,}2{1,,`1{1`,,2`,,}2,,}2,,} has recursion level C(C(Ω22+C(Ω2+ω^(σ+++C(C(Ω22,σ),σ+))+C(C(Ω2+ω^(σ+++C(C(Ω22,σ),σ+))+1,σ),C(C(Ω22,σ),σ+)),0),0),0)
  • {1{1,,`1{1{1{1,,`1{1`,,2`,,}2,,}2,,}2`,,}1{1{1,,`1{2`,,}2,,}2,,}2,,}2{1,,`1{1`,,2`,,}2,,}2,,} has recursion level C(C(Ω22+C(Ω2+ω^(σ+++C(C(Ω22,σ),σ+))+C(C(Ω2+ω^(σ+++C(C(Ω22,σ),σ+))+C(C(Ω2+ω^(σ+++1),σ),σ+),σ),C(C(Ω22,σ),σ+)),0),0),0)
  • {1{1,,`1{1{1{1,,`1{1`,,2`,,}2,,}2,,}2`,,}1{1{1,,`1{1`,,2`,,}2,,}2,,}2,,}2{1,,`1{1`,,2`,,}2,,}2,,} has recursion level C(C(Ω22+C(Ω2+ω^(σ+++C(C(Ω22,σ),σ+))+C(C(Ω2+ω^(σ+++C(C(Ω22,σ),σ+))+C(C(Ω22,σ),σ+),σ),C(C(Ω22,σ),σ+)),0),0),0)
  • {1{1,,`1{1{1{1,,`1{1`,,2`,,}2,,}2,,}2`,,}1{2{1,,`1{1`,,2`,,}2,,}2,,}2,,}2{1,,`1{1`,,2`,,}2,,}2,,} has recursion level C(C(Ω22+C(Ω2+ω^(σ+++C(C(Ω22,σ),σ+))+C(C(Ω2+ω^(σ+++C(C(Ω22,σ),σ+))+ω^(C(C(Ω22,σ),σ+)+1),σ),C(C(Ω22,σ),σ+)),0),0),0)
  • {1{1,,`1{1{1{1,,`1{1`,,2`,,}2,,}2,,}2`,,}1`,,2,,}2{1,,`1{1`,,2`,,}2,,}2,,} has recursion level C(C(Ω22+C(Ω2+ω^(σ+++C(C(Ω22,σ),σ+))+C(C(Ω22,σ),C(C(Ω22,σ),σ+)),0),0),0)
  • {1{1,,`1{1{1{1,,`1{1`,,2`,,}2,,}2,,}2`,,}1`,,2,,}1,2{1,,`1{1`,,2`,,}2,,}2,,} has recursion level C(C(Ω22+C(Ω2+ω^(σ+++C(C(Ω22,σ),σ+))+C(C(Ω22,σ)+1,σ+),0),0),0)
  • {1{1{1,,`1,2,,}2,,`1{1{1{1,,`1{1`,,2`,,}2,,}2,,}2`,,}1`,,2,,}2{1,,`1{1`,,2`,,}2,,}2,,} has recursion level C(C(Ω22+C(Ω2+ω^(σ+++C(C(Ω22,σ),σ+))+C(C(Ω22,σ)+C(Ω2+1,σ),σ+),0),0),0)
  • {1{1{1,,`1{1{1{1,,`1{1`,,2`,,}2,,}2,,}2`,,}2,,}2,,`1{1{1{1,,`1{1`,,2`,,}2,,}2,,}2`,,}1`,,2,,} 2{1,,`1{1`,,2`,,}2,,}2,,} has recursion level C(C(Ω22+C(Ω2+ω^(σ+++C(C(Ω22,σ),σ+))+C(C(Ω22,σ)+C(Ω2+ω^(σ+++C(C(Ω22,σ),σ+)),σ),σ+),0),0),0)
  • {1{1{1,,`1{1{1{1,,`1{1`,,2`,,}2,,}2,,}2`,,}1`,,2,,}2,,`1{1{1{1,,`1{1`,,2`,,}2,,}2,,}2`,,}1`,,2,,} 2{1,,`1{1`,,2`,,}2,,}2,,} has recursion level C(C(Ω22+C(Ω2+ω^(σ+++C(C(Ω22,σ),σ+))+C(C(Ω22,σ)2,σ+),0),0),0)
  • {1{1,,`2{1{1{1,,`1{1`,,2`,,}2,,}2,,}2`,,}1`,,2,,}2{1,,`1{1`,,2`,,}2,,}2,,} has recursion level C(C(Ω22+C(Ω2+ω^(σ+++C(C(Ω22,σ),σ+))+C(C(Ω22,σ)++),0),0),0)
  • {1{1,,`1{1{1{1,,`1{1`,,2`,,}2,,}2,,}2`,,}2`,,2,,}2{1,,`1{1`,,2`,,}2,,}2,,} has recursion level C(C(Ω22+C(Ω2+ω^(σ+++C(C(Ω22,σ),σ+))+C(C(Ω2+ω^(σ+++C(C(Ω22,σ),σ+)),C(Ω22,σ)),σ+),0),0),0)
  • {1{1,,`1{1{1{1,,`1{1`,,2`,,}2,,}2,,}2`,,}1`,,3,,}2{1,,`1{1`,,2`,,}2,,}2,,} has recursion level C(C(Ω22+C(Ω2+ω^(σ+++C(C(Ω22,σ),σ+))+C(C(Ω22,C(Ω22,σ)),σ+),0),0),0)
  • {1{1,,`1{1{1{1,,`1{1`,,2`,,}2,,}2,,}2`,,}1`,,1,2,,}2{1,,`1{1`,,2`,,}2,,}2,,} has recursion level C(C(Ω22+C(Ω2+ω^(σ+++C(C(Ω22,σ),σ+))+C(C(Ω22+1,0),σ+),0),0),0)
  • {1{1,,`1{1{1{1,,`1{1`,,2`,,}2,,}2,,}2`,,}1`,,1,,2,,}2{1,,`1{1`,,2`,,}2,,}2,,} has recursion level C(C(Ω22+C(Ω2+ω^(σ+++C(C(Ω22,σ),σ+))+σ++,0),0),0)
  • {1{1,,`1{1{1{1,,`1{1`,,2`,,}2,,}2,,}2`,,}1{2`,,}2,,}2{1,,`1{1`,,2`,,}2,,}2,,} has recursion level C(C(Ω22+C(Ω2+ω^(σ+++C(C(Ω22,σ),σ+))+ω^(σ+++1),0),0),0)
  • {1{1,,`1{1{1{1,,`1{1`,,2`,,}2,,}2,,}2`,,}1{1{1{1,,`1{1`,,2`,,}2,,}2,,}2`,,}2,,}2{1,,`1{1`,,2`,,}2,,}2,,} has recursion level C(C(Ω22+C(Ω2+ω^(σ+++C(C(Ω22,σ),σ+))2,0),0),0)
  • {1{1,,`1{2{1{1,,`1{1`,,2`,,}2,,}2,,}2`,,}2,,}2{1,,`1{1`,,2`,,}2,,}2,,} has recursion level C(C(Ω22+C(Ω2+ω^(σ+++C(C(Ω22,σ),σ+)+1),0),0),0)
  • {1{1,,`1{1{1{1,,`1{1`,,2`,,}2,,}2,,}3`,,}2,,}2{1,,`1{1`,,2`,,}2,,}2,,} has recursion level C(C(Ω22+C(Ω2+ω^(σ+++C(C(Ω22,σ),σ+)2),0),0),0)
  • {1{1,,`1{1{1{1,,`1{2`,,}2,,}2{1,,`1{1`,,2`,,}2,,}2,,}2`,,}2,,}2{1,,`1{1`,,2`,,}2,,}2,,} has recursion level C(C(Ω22+C(Ω2+ω^(σ+++C(C(Ω2+ω^(σ+++1),σ),C(C(Ω22,σ),σ+))),0),0),0)
  • {1{1,,`1{1`,,2`,,}2,,}3,,} has recursion level C(C(Ω22+C(Ω2+ω^(σ+++C(C(Ω22,σ),C(C(Ω22,σ),σ+))),0),0),0) (now it returns to the normal)
  • {1{1,,`1{1`,,2`,,}2,,}1,2,,} has recursion level C(C(Ω22+C(Ω2+ω^(σ+++C(C(Ω22,σ)+1,σ+)),0),0),0)
  • {1,,`2{1`,,2`,,}2,,} has recursion level C(C(Ω22+C(Ω2+ω^(σ+++C(C(Ω22,σ)++)),0),0),0)
  • {1,,`1{1`,,2`,,}1,2,,} has recursion level C(C(Ω22+C(Ω2+ω^(σ+++C(C(Ω22+1,0),σ+)),0),0),0)
  • {1,,`1{1`,,2`,,}1,,2,,} has recursion level C(C(Ω22+C(Ω2+ω^(σ++2),0),0),0)
  • {1,,`1{1`,,2`,,}1{2`,,}2,,} has recursion level C(C(Ω22+C(Ω2+ω^(σ++2)+ω^(σ+++1),0),0),0)
  • {1,,`1{1`,,2`,,}1{1`,,2`,,}1,,2,,} has recursion level C(C(Ω22+C(Ω2+ω^(σ++2)2,0),0),0)
  • {1,,`1{2`,,2`,,}2,,} has recursion level C(C(Ω22+C(Ω2+ω^(σ++2+1),0),0),0)
  • {1,,`1{1`,,1,2`,,}2,,} has recursion level C(C(Ω22+C(Ω2+ω^ω^(σ+++1),0),0),0)
  • {1,,`1{1` `,,2`,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(σ+++++),0),0),0)
  • {1,,`1{1{1,,`1,2,,}2`,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(C(Ω2+1,σ),σ++),0),0),0)
  • {1,,`1{1{1,,`1,,1,2,,}2`,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(C(Ω2++1,σ),σ++),0),0),0)
  • {1,,`1{1{1,,`1{1{1,,`1,2,,}2,,}2,,}2`,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(C(Ω2+C(C(Ω2+1,σ),σ+),σ),σ++),0),0),0)
  • {1,,`1{1{1,,`1{1{1,,`1`,,2,,}2,,}2,,}2`,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(C(Ω2+C(C(Ω2++,σ),σ+),σ),σ++),0),0),0)
  • {1,,`1{1{1,,`1{1{1,,`1{1{1,,`1,2,,}2`,,}2,,}2,,}2,,}2`,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(C(Ω2+C(C(Ω2+C(C(Ω2+1,σ),σ++),σ),σ+),σ),σ++),0),0),0)
  • {1,,`1{1{1,,`1`,,2,,}2`,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(C(Ω2+C(C(Ω22,σ),σ+),σ),σ++),0),0),0)
  • {1,,`2{1{1,,`1`,,2,,}2`,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(C(Ω2+C(C(Ω22,σ)++),σ),σ++),0),0),0)
  • {1,,`1{1{1,,`1`,,2,,}2`,,}1,2,,} has recursion level C(C(Ω22+C(Ω2+C(C(Ω2+C(C(Ω22+1,0),σ+),σ),σ++),0),0),0)
  • {1,,`1{1{1,,`1`,,2,,}2`,,}1,,2,,} has recursion level C(C(Ω22+C(Ω2+C(C(Ω2++,σ),σ++),0),0),0)
  • {1,,`1{2{1,,`1`,,2,,}2`,,}2,,} has recursion level C(C(Ω22+C(Ω2+ω^(C(C(Ω2++,σ),σ++)+1),0),0),0)
  • {1,,`1{1{1,,`1`,,2,,}1,2`,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(C(Ω2++,σ)+1,σ++),0),0),0)
  • {1,,`1{1{1,,`2`,,2,,}2`,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(C(Ω2++,σ)+++),0),0),0)
  • {1,,`1{1{1,,`1`,,1,2,,}2`,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(C(Ω2+++1,σ),σ++),0),0),0)
  • {1,,`1{1{1,,`1{1{1,,`1,2,,}2`,,}2,,}2`,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(C(Ω2+C(C(Ω2+1,σ),σ++),σ),σ++),0),0),0)
  • {1,,`1{1{1,,`1{1{1,,`1`,,2,,}2`,,}2,,}2`,,}1,,2,,} has recursion level C(C(Ω22+C(Ω2+C(C(Ω2+C(C(Ω2++,σ),σ++),σ),σ++),0),0),0)
  • {1,,`1` `,,2,,} has recursion level C(C(Ω22+C(Ω2+C(C(Ω22,σ),σ++),0),0),0)
  • {1{1,,`1` `,,2,,}1,2} has recursion level C(C(Ω22+C(Ω2+C(C(Ω22,σ),σ++)+1,0),0),0)
  • {1{1,,`1` `,,2,,}1{1,,`1`,,2,,}2} has recursion level C(C(Ω22+C(Ω2+C(C(Ω22,σ),σ++)+C(C(Ω2++,σ),σ+),0),0),0)
  • {1{1,,`1` `,,2,,}1{1,,`1{1{1,,`1`,,2,,}2`,,}2,,}2} has recursion level C(C(Ω22+C(Ω2+C(C(Ω22,σ),σ++)+C(C(Ω2+C(C(Ω2++,σ),σ++),σ),σ+),0),0),0)
  • {1{1,,`1` `,,2,,}1{1,,`1` `,,2,,}2} has recursion level C(C(Ω22+C(Ω2+C(C(Ω22,σ),σ++)+C(C(Ω22,σ),σ+),0),0),0) (here’s another similar yet erratic thing)
  • {1{1,,`1`,,2,,}2{1,,`1` `,,2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(C(Ω22,σ),σ++)+C(C(Ω2++,σ),C(C(Ω22,σ),σ+)),0),0),0)
  • {1{1,,`1` `,,2,,}3,,} has recursion level C(C(Ω22+C(Ω2+C(C(Ω22,σ),σ++)+C(C(Ω22,σ),C(C(Ω22,σ),σ+)),0),0),0)
  • {1{1,,`1` `,,2,,}1,2,,} has recursion level C(C(Ω22+C(Ω2+C(C(Ω22,σ),σ++)+C(C(Ω22,σ)+1,σ+),0),0),0)
  • {1,,`2` `,,2,,} has recursion level C(C(Ω22+C(Ω2+C(C(Ω22,σ),σ++)+C(C(Ω22,σ)++),0),0),0)
  • {1,,`1`,,2` `,,2,,} has recursion level C(C(Ω22+C(Ω2+C(C(Ω22,σ),σ++)+C(C(Ω2++,C(Ω22,σ)),σ+),0),0),0)
  • {1,,`1{1{1,,`1` `,,2,,}2`,,}2` `,,2,,} has recursion level C(C(Ω22+C(Ω2+C(C(Ω22,σ),σ++)+C(C(Ω22,C(Ω22,σ)),σ+),0),0),0)
  • {1,,`1{1{1,,`1` `,,2,,}2`,,}1,2` `,,2,,} has recursion level C(C(Ω22+C(Ω2+C(C(Ω22,σ),σ++)+C(C(Ω22+1,0),σ+),0),0),0)
  • {1,,`1{1{1,,`1` `,,2,,}2`,,}1,,2` `,,2,,} has recursion level C(C(Ω22+C(Ω2+C(C(Ω22,σ),σ++)+σ++,0),0),0)
  • {1{1,,`1{1{1,,`1` `,,2,,}2`,,}1,,2` `,,2,,}1,2} has recursion level C(C(Ω22+C(Ω2+C(C(Ω22,σ),σ++)+σ+++1,0),0),0)
  • {1{1,,`1{1{1,,`1` `,,2,,}2`,,}1,,2` `,,2,,}1{1,,`1`,,2,,}2} has recursion level C(C(Ω22+C(Ω2+C(C(Ω22,σ),σ++)+σ+++C(C(Ω2++,σ),σ+),0),0),0)
  • {1{1,,`1{1{1,,`1` `,,2,,}2`,,}1,,2` `,,2,,}1{1,,`1` `,,2,,}2} has recursion level C(C(Ω22+C(Ω2+C(C(Ω22,σ),σ++)+σ+++C(C(Ω22,σ),σ+),0),0),0)
  • {1{1,,`1{1{1,,`1` `,,2,,}2`,,}1,,2` `,,2,,}1{1,,`1{1{1,,`1` `,,2,,}2`,,}1,2` `,,2,,}2} has recursion level C(C(Ω22+C(Ω2+C(C(Ω22,σ),σ++)+σ+++C(C(Ω22+1,0),σ+),0),0),0)
  • {1{1,,`1{1{1,,`1` `,,2,,}2`,,}1,,2` `,,2,,}1{1,,`1{1{1,,`1` `,,2,,}2`,,}1,,2` `,,2,,}2} has recursion level C(C(Ω22+C(Ω2+C(C(Ω22,σ),σ++)+σ++2,0),0),0)
  • {1{2{1,,`1{1{1,,`1` `,,2,,}2`,,}1,,2` `,,2,,}2,,}2} has recursion level C(C(Ω22+C(Ω2+C(C(Ω22,σ),σ++)+ω^(σ+++1),0),0),0)
  • {1{1,,`1,2,,}2{1,,`1{1{1,,`1` `,,2,,}2`,,}1,,2` `,,2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(C(Ω22,σ),σ++)+C(C(Ω2+1,σ),σ++),0),0),0)
  • {1{1,,`1{1{1,,`1` `,,2,,}2,,}2,,}2{1,,`1{1{1,,`1` `,,2,,}2`,,}1,,2` `,,2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(C(Ω22,σ),σ++)+C(C(Ω2+C(C(Ω22,σ),σ+),σ),σ++),0),0),0)
  • {1{1,,`1{1{1,,`1{1{1,,`1` `,,2,,}2`,,}1,2` `,,2,,}2,,}2,,}2{1,,`1{1{1,,`1` `,,2,,}2`,,}1,,2` `,,2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(C(Ω22,σ),σ++)+C(C(Ω2+C(C(Ω22+1,0),σ+),σ),σ++),0),0),0)
  • {1{1,,`1{1{1,,`1{1{1,,`1` `,,2,,}2`,,}1,,2` `,,2,,}2,,}2,,}2{1,,`1{1{1,,`1` `,,2,,}2`,,}1,,2` `,,2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(C(Ω22,σ),σ++)+C(C(Ω2++,σ),σ++),0),0),0)
  • {1{1,,`2{1{1,,`1{1{1,,`1` `,,2,,}2`,,}1,,2` `,,2,,}2,,}2,,}2{1,,`1{1{1,,`1` `,,2,,}2`,,}1,,2` `,,2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(C(Ω22,σ),σ++)+C(C(Ω2++,σ)+++),0),0),0)
  • {1{1,,`1{1{1,,`1{1{1,,`1` `,,2,,}2`,,}1,,2` `,,2,,}2,,}1,2,,}2{1,,`1{1{1,,`1` `,,2,,}2`,,}1,,2` `,,2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(C(Ω22,σ),σ++)+C(C(Ω2+++1,σ),σ++),0),0),0)
  • {1{1,,`1{1{1,,`1,2,,}2{1,,`1{1{1,,`1` `,,2,,}2`,,}1,,2` `,,2,,}2,,}2,,}2{1,,`1{1{1,,`1` `,,2,,}2`,,}1,,2` `,,2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(C(Ω22,σ),σ++)+C(C(Ω2+C(C(Ω2+1,σ),σ++),σ),σ++),0),0),0)
  • {1{1,,`1`,,2,,}2{1,,`1{1{1,,`1` `,,2,,}2`,,}1,,2` `,,2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(C(Ω22,σ),σ++)2,0),0),0)
  • {1{1{1,,`1`,,2,,}2{1,,`1{1{1,,`1` `,,2,,}2`,,}1,,2` `,,2,,}2,,}1,2} has recursion level C(C(Ω22+C(Ω2+C(C(Ω22,σ),σ++)2+1,0),0),0)
  • {1{1{1,,`1`,,2,,}2{1,,`1{1{1,,`1` `,,2,,}2`,,}1,,2` `,,2,,}2,,}1{1{1,,`1` `,,2,,}2,,}2} has recursion level C(C(Ω22+C(Ω2+C(C(Ω22,σ),σ++)2+C(C(Ω22,σ),σ+),0),0),0)
  • {1{1{1,,`1`,,2,,}2{1,,`1{1{1,,`1` `,,2,,}2`,,}1,,2` `,,2,,}2,,}1{1{1,,`1{1{1,,`1` `,,2,,}2`,,}1,,2` `,,2,,}2,,}2} has recursion level C(C(Ω22+C(Ω2+C(C(Ω22,σ),σ++)2+σ++,0),0),0)
  • {1{1{1,,`1`,,2,,}2{1,,`1{1{1,,`1` `,,2,,}2`,,}1,,2` `,,2,,}2,,}1{1{1,,`1`,,2,,}2{1,,`1{1{1,,`1` `,,2,,}2`,,}1,,2` `,,2,,}2,,}2} has recursion level C(C(Ω22+C(Ω2+C(C(Ω22,σ),σ++)3,0),0),0)
  • {1{2{1,,`1`,,2,,}2{1,,`1{1{1,,`1` `,,2,,}2`,,}1,,2` `,,2,,}2,,}2} has recursion level C(C(Ω22+C(Ω2+ω^(C(C(Ω22,σ),σ++)+1),0),0),0)
  • {1{1,,`1,2,,}2{1,,`1`,,2,,}2{1,,`1{1{1,,`1` `,,2,,}2`,,}1,,2` `,,2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(C(Ω2+1,σ),C(C(Ω22,σ),σ++)),0),0),0)
  • {1{1,,`1{1{1,,`1` `,,2,,}2,,}2,,}2{1,,`1`,,2,,}2{1,,`1{1{1,,`1` `,,2,,}2`,,}1,,2` `,,2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(C(Ω2+C(C(Ω22,σ),σ+),σ),C(C(Ω22,σ),σ++)),0),0),0)
  • {1{1,,`1{1{1,,`1{1{1,,`1` `,,2,,}2`,,}1,,2` `,,2,,}2,,}2,,}2{1,,`1`,,2,,}2{1,,`1{1{1,,`1` `,,2,,}2`,,}1,,2` `,,2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(C(Ω2++,σ),C(C(Ω22,σ),σ++)),0),0),0)
  • {1{1,,`1{1{1,,`1,2,,}2{1,,`1{1{1,,`1` `,,2,,}2`,,}1,,2` `,,2,,}2,,}2,,}2{1,,`1`,,2,,}2{1,,`1{1{1,,`1` `,,2,,}2`,,}1,,2` `,,2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(C(Ω2+C(C(Ω2+1,σ),σ++),σ),C(C(Ω22,σ),σ++)),0),0),0)
  • {1{1,,`1{1{1,,`1{1{1,,`1{1{1,,`1` `,,2,,}2`,,}1,,2` `,,2,,}2,,}2,,}2{1,,`1{1{1,,`1` `,,2,,}2`,,}1,,2` `,,2,,}2,,}2,,}2{1,,`1`,,2,,}2{1,,`1{1{1,,`1` `,,2,,}2`,,}1,,2` `,,2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(C(Ω2+C(C(Ω2++,σ),σ++),σ),C(C(Ω22,σ),σ++)),0),0),0)
  • {1{1,,`1{1{1,,`1`,,2,,}2{1,,`1{1{1,,`1` `,,2,,}2`,,}1,,2` `,,2,,}2,,}2,,}2{1,,`1`,,2,,}2{1,,`1{1{1,,`1` `,,2,,}2`,,}1,,2` `,,2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(C(Ω2+C(C(Ω22,σ),σ++),σ),C(C(Ω22,σ),σ++)),0),0),0)
  • {1{1,,`1{1{1,,`1,2,,}2{1,,`1`,,2,,}2{1,,`1{1{1,,`1` `,,2,,}2`,,}1,,2` `,,2,,}2,,}2,,}2{1,,`1`,,2,,}2{1,,`1{1{1,,`1` `,,2,,}2`,,}1,,2` `,,2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(C(Ω2+C(C(Ω2+1,σ),C(C(Ω22,σ),σ++)),σ),C(C(Ω22,σ),σ++)),0),0),0)
  • {1{1,,`1`,,2,,}3{1,,`1{1{1,,`1` `,,2,,}2`,,}1,,2` `,,2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(C(Ω22,σ),C(C(Ω22,σ),σ++)),0),0),0)
  • {1{1,,`1`,,2,,}1,2{1,,`1{1{1,,`1` `,,2,,}2`,,}1,,2` `,,2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(C(Ω22,σ)+1,σ++),0),0),0)
  • {1{1{1,,`1,2,,}2,,`1`,,2,,}2{1,,`1{1{1,,`1` `,,2,,}2`,,}1,,2` `,,2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(C(Ω22,σ)+C(Ω2+1,σ),σ++),0),0),0)
  • {1{1{1,,`1{1{1,,`1{1{1,,`1` `,,2,,}2`,,}1,,2` `,,2,,}2,,}2,,}2,,`1`,,2,,}2{1,,`1{1{1,,`1` `,,2,,}2`,,}1,,2` `,,2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(C(Ω22,σ)+C(Ω2++,σ),σ++),0),0),0)
  • {1{1{1,,`1`,,2,,}2,,`1`,,2,,}2{1,,`1{1{1,,`1` `,,2,,}2`,,}1,,2` `,,2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(C(Ω22,σ)2,σ++),0),0),0)
  • {1{1,,`2`,,2,,}2{1,,`1{1{1,,`1` `,,2,,}2`,,}1,,2` `,,2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(C(Ω22,σ)+++),0),0),0)
  • {1{1,,`1{1{1,,`1{1{1,,`1` `,,2,,}2`,,}1,,2` `,,2,,}2,,}2`,,2,,}2{1,,`1{1{1,,`1` `,,2,,}2`,,}1,,2` `,,2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(C(Ω2++,C(Ω22,σ)),σ++),0),0),0)
  • {1{1,,`1`,,3,,}2{1,,`1{1{1,,`1` `,,2,,}2`,,}1,,2` `,,2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(C(Ω22,C(Ω22,σ)),σ++),0),0),0)
  • {1{1,,`1`,,1,2,,}2{1,,`1{1{1,,`1` `,,2,,}2`,,}1,,2` `,,2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(C(Ω22+1,0),σ++),0),0),0)
  • {1{1,,`1`,,1{1{1{1,,`1`,,1,2,,}2{1,,`1{1{1,,`1` `,,2,,}2`,,}1,,2` `,,2,,}2,,}2}2,,}2{1,,`1{1{1,,`1` `,,2,,}2`,,}1,,2` `,,2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(C(Ω22+C(Ω2+C(C(Ω22+1,0),σ++),0),0),σ++),0),0),0)
  • {1{1,,`1`,,1,,2,,}2{1,,`1{1{1,,`1` `,,2,,}2`,,}1,,2` `,,2,,}2,,} has recursion level C(C(Ω22+C(Ω2+++,0),0),0)

Up to here, it seems that C(Ω2,C(Ω22,0)) corresponds to the double comma, and C(Ω2,C(Ω2,C(Ω22,0))) corresponds to {1{1,,`1{1{1,,`1` `,,2,,}2`,,}1,,2` `,,2,,}2,,}.

Up to a C(Ω2+C(Ω22,0),C(Ω22,0))

Now let separator ■ = {1,,`1{1{1,,`1` `,,2,,}2`,,}1,,2` `,,2,,} and separator ♦ = {1{1{1,,`1`,,1,,2,,}2■2,,}2} = {1{1{1,,`1`,,1,,2,,}2{1,,`1{1{1,,`1` `,,2,,}2`,,}1,,2` `,,2,,}2,,}2} and let ordinal σ = C(Ω22,0). And α+ is shorthand for C(Ω2,α).

  • {1{1{1,,`1`,,1♦2,,}2■2,,}2} has recursion level C(C(Ω22+C(Ω2+++,0),0),0)
  • {1{1,,1,,1,2}2{1{1,,`1`,,1♦2,,}2■2,,}2} has recursion level C(C(Ω22+C(Ω2+++,0)+C(Ω2++1,0),0),0)
  • {1{1{1,,`1`,,1{1{1,,`1`,,1,,2,,}2}2,,}2}2{1{1,,`1`,,1♦2,,}2■2,,}2} has recursion level C(C(Ω22+C(Ω2+++,0)+C(Ω2+C(C(Ω22+C(Ω2++,0),0),σ+),0),0),0)
  • {1{1{1,,`1`,,1{1{1{1,,`1`,,1♦2,,}2■2,,}2}2,,}2}2{1{1,,`1`,,1♦2,,}2■2,,}2} has recursion level C(C(Ω22+C(Ω2+++,0)+C(Ω2+C(C(Ω22+C(Ω2+++,0),0),σ+),0),0),0)
  • {1{1{1,,`1`,,1,,2,,}2}2{1{1,,`1`,,1♦2,,}2■2,,}2} has recursion level C(C(Ω22+C(Ω2+++,0)+C(Ω2++,0),0),0)
  • {1{1{1,,`1` `,,2,,}2}2{1{1,,`1`,,1♦2,,}2■2,,}2} has recursion level C(C(Ω22+C(Ω2+++,0)+C(Ω2+C(C(Ω22,σ),σ++),0),0),0)
  • {1{1{1{1,,`1`,,1♦2,,}2■2,,}2}2{1{1,,`1`,,1♦2,,}2■2,,}2} has recursion level C(C(Ω22+C(Ω2+++,0)+C(Ω2+C(C(Ω22+C(Ω2+++,0),0),σ++),0),0),0)
  • {1,,1,,2{1{1,,`1`,,1♦2,,}2■2,,}2} has recursion level C(C(Ω22+C(Ω2+++,0)+C(σ,C(Ω2+C(C(Ω22+C(Ω2+++,0),0),σ++),0)),0),0)
  • {1,,1,,1,2{1{1,,`1`,,1♦2,,}2■2,,}2} has recursion level C(C(Ω22+C(Ω2+++,0)+C(Ω2+C(C(Ω22+C(Ω2++1,C(Ω2+C(C(Ω22+C(Ω2+++,0),0),σ++),0)),0),σ),C(Ω2+C(C(Ω22+C(Ω2+++,0),0),σ++),0)),0),0)
  • {1{1,,`1`,,1,,2,,}2{1{1,,`1`,,1♦2,,}2■2,,}2} has recursion level C(C(Ω22+C(Ω2+++,0)+C(Ω2+C(C(Ω22+C(Ω2++,C(Ω2+C(C(Ω22+C(Ω2+++,0),0),σ++),0)),0),σ),C(Ω2+C(C(Ω22+C(Ω2+++,0),0),σ++),0)),0),0)
  • {1{1{1,,`1`,,1♦2,,}2■2,,}3} has recursion level C(C(Ω22+C(Ω2+++,0)+C(Ω2+C(C(Ω22+C(Ω2+++,0),0),σ),C(Ω2+C(C(Ω22+C(Ω2+++,0),0),σ++),0)),0),0)
  • {1{1{1{1,,`1`,,1♦2,,}2■2,,}3}2{1{1,,`1`,,1♦2,,}2■2,,}3} has recursion level C(C(Ω22+C(Ω2+++,0)+C(Ω2+C(C(Ω22+C(Ω2+++,0),0),σ++),C(Ω2+C(C(Ω22+C(Ω2+++,0),0),σ++),0)),0),0)
  • {1{1{1,,`1`,,1♦2,,}2■2,,}1,2} has recursion level C(C(Ω22+C(Ω2+++,0)+C(Ω2+C(C(Ω22+C(Ω2+++,0),0),σ++)+1,0),0),0)
  • {1{1{1,,`1`,,1♦2,,}2■2,,}1{1,,`1` `,,2,,}2} has recursion level C(C(Ω22+C(Ω2+++,0)+C(Ω2+C(C(Ω22+C(Ω2+++,0),0),σ++)+C(C(Ω22,σ),σ+),0),0),0)
  • {1{1{1,,`1`,,1♦2,,}2■2,,}1{1,,`1{1{1,,`1` `,,2,,}2`,,}1,2` `,,2,,}2} has recursion level C(C(Ω22+C(Ω2+++,0)+C(Ω2+C(C(Ω22+C(Ω2+++,0),0),σ++)+C(C(Ω22+1,0),σ+),0),0),0)
  • {1{1{1,,`1`,,1♦2,,}2■2,,}1{1,,`1{1{1,,`1` `,,2,,}2`,,}1{1{1{1,,`1`,,1♦2,,}2■2,,}2}2` `,,2,,}2} has recursion level C(C(Ω22+C(Ω2+++,0)+C(Ω2+C(C(Ω22+C(Ω2+++,0),0),σ++)+C(C(Ω22+C(Ω2+++,0),0),σ+),0),0),0)
  • {1{1{1,,`1`,,1♦2,,}2■2,,}1■2} has recursion level C(C(Ω22+C(Ω2+++,0)+C(Ω2+C(C(Ω22+C(Ω2+++,0),0),σ++)+σ++,0),0),0)
  • {1{1{1,,`1`,,1♦2,,}2■2,,}1{1{1,,`1`,,2,,}2■2,,}2} has recursion level C(C(Ω22+C(Ω2+++,0)+C(Ω2+C(C(Ω22+C(Ω2+++,0),0),σ++)+C(C(Ω22,σ),σ++),0),0),0)
  • {1{1{1,,`1`,,1♦2,,}2■2,,}1{1{1,,`1`,,2,,}1,2■2,,}2} has recursion level C(C(Ω22+C(Ω2+++,0)+C(Ω2+C(C(Ω22+C(Ω2+++,0),0),σ++)+C(C(Ω22,σ)+1,σ++),0),0),0)
  • {1{1{1,,`1`,,1♦2,,}2■2,,}1{1{1,,`1`,,3,,}2■2,,}2} has recursion level C(C(Ω22+C(Ω2+++,0)+C(Ω2+C(C(Ω22+C(Ω2+++,0),0),σ++)+C(C(Ω22,C(Ω22,σ)),σ++),0),0),0)
  • {1{1{1,,`1`,,1♦2,,}2■2,,}1{1{1,,`1`,,1,2,,}2■2,,}2} has recursion level C(C(Ω22+C(Ω2+++,0)+C(Ω2+C(C(Ω22+C(Ω2+++,0),0),σ++)+C(C(Ω22+1,0),σ++),0),0),0)
  • {1{1{1,,`1`,,1♦2,,}2■2,,}1{1{1,,`1`,,1♦2,,}2■2,,}2} has recursion level C(C(Ω22+C(Ω2+++,0)+C(Ω2+C(C(Ω22+C(Ω2+++,0),0),σ++)2,0),0),0)
  • {1{2{1,,`1`,,1♦2,,}2■2,,}2} has recursion level C(C(Ω22+C(Ω2+++,0)+C(Ω2+ω^(C(C(Ω22+C(Ω2+++,0),0),σ++)+1),0),0),0)
  • {1{1,,`1,2,,}2{1,,`1`,,1♦2,,}2■2,,} has recursion level C(C(Ω22+C(Ω2+++,0)+C(Ω2+C(C(Ω2+1,σ),C(C(Ω22+C(Ω2+++,0),0),σ++)),0),0),0)
  • {1{1,,`1{1■2,,}2,,}2{1,,`1`,,1♦2,,}2■2,,} has recursion level C(C(Ω22+C(Ω2+++,0)+C(Ω2+C(C(Ω2++,σ),C(C(Ω22+C(Ω2+++,0),0),σ++)),0),0),0)
  • {1{1,,`1{1{1,,`1`,,1♦2,,}2■2,,}2,,}2{1,,`1`,,1♦2,,}2■2,,} has recursion level C(C(Ω22+C(Ω2+++,0)+C(Ω2+C(C(Ω2+C(C(Ω22+C(Ω2+++,0),0),σ++),σ),C(C(Ω22+C(Ω2+++,0),0),σ++)),0),0),0)
  • {1{1,,`1`,,2,,}2{1,,`1`,,1♦2,,}2■2,,} has recursion level C(C(Ω22+C(Ω2+++,0)+C(Ω2+C(C(Ω22,σ),C(C(Ω22+C(Ω2+++,0),0),σ++)),0),0),0)
  • {1{1,,`1`,,2,,}1,2{1,,`1`,,1♦2,,}2■2,,} has recursion level C(C(Ω22+C(Ω2+++,0)+C(Ω2+C(C(Ω22,σ)+1,C(C(Ω22+C(Ω2+++,0),0),σ++)),0),0),0)
  • {1{1,,`2`,,2,,}2{1,,`1`,,1♦2,,}2■2,,} has recursion level C(C(Ω22+C(Ω2+++,0)+C(Ω2+C(C(Ω22,σ)+,C(C(Ω22+C(Ω2+++,0),0),σ++)),0),0),0)
  • {1{1,,`1`,,3,,}2{1,,`1`,,1♦2,,}2■2,,} has recursion level C(C(Ω22+C(Ω2+++,0)+C(Ω2+C(C(Ω22,C(Ω22,σ)),C(C(Ω22+C(Ω2+++,0),0),σ++)),0),0),0)
  • {1{1,,`1`,,1,2,,}2{1,,`1`,,1♦2,,}2■2,,} has recursion level C(C(Ω22+C(Ω2+++,0)+C(Ω2+C(C(Ω22+1,0),C(C(Ω22+C(Ω2+++,0),0),σ++)),0),0),0)
  • {1{1,,`1`,,1♦2,,}3■2,,} has recursion level C(C(Ω22+C(Ω2+++,0)+C(Ω2+C(C(Ω22+C(Ω2+++,0),0),C(C(Ω22+C(Ω2+++,0),0),σ++)),0),0),0)
  • {1{1,,`1`,,1♦2,,}1,2■2,,} has recursion level C(C(Ω22+C(Ω2+++,0)+C(Ω2+C(C(Ω22+C(Ω2+++,0),0)+1,σ++),0),0),0)
  • {1{1,,`2`,,1♦2,,}2■2,,} has recursion level C(C(Ω22+C(Ω2+++,0)+C(Ω2+C(C(Ω22+C(Ω2+++,0),0)+++),0),0),0)
  • {1{1,,`1`,,1♦1,2,,}2■2,,} has recursion level C(C(Ω22+C(Ω2+++,0)+C(Ω2+C(C(Ω22+C(Ω2+++,0)+1,0),σ++),0),0),0)
  • {1{1,,`1`,,1♦1♦2,,}2■2,,} has recursion level C(C(Ω22+C(Ω2+++,0)2,0),0)
  • {1{1,,`1`,,1,,2,,}2{1{1,,`1`,,1,,2,,}2■2,,}2} has recursion level C(C(Ω22+C(Ω2++,C(Ω2+++,0)),0),0)
  • {1{1{1,,`1`,,1{1{1{1,,`1`,,1♦2,,}2■2,,}2}2,,}2■2,,}2{1{1,,`1`,,1,,2,,}2■2,,}2} has recursion level C(C(Ω22+C(Ω2+C(C(Ω22+C(Ω2+C(C(Ω22+C(Ω2+++,0),0),σ++),0),0),σ++),C(Ω2+++,0)),0),0)
  • {1{1{1,,`1`,,1♦2,,}2■2,,}2{1{1,,`1`,,1,,2,,}2■2,,}2} has recursion level C(C(Ω22+C(Ω2+C(C(Ω22+C(Ω2+++,0),0),σ++),C(Ω2+++,0)),0),0)
  • {1{1{1,,`1`,,1,,2,,}2■2,,}3} has recursion level C(C(Ω22+C(Ω2+++,C(Ω2+++,0)),0),0)
  • {1{1{1,,`1`,,1,,2,,}2■2,,}1,2} has recursion level C(C(Ω22+C(Ω2++++1,0),0),0)
  • {1{1{1,,`1`,,1,,2,,}2■2,,}1{1,,`1,2,,}2} has recursion level C(C(Ω22+C(Ω2++++C(C(Ω2+1,σ),σ+),0),0),0)
  • {1{1{1,,`1`,,1,,2,,}2■2,,}1{1,,`1`,,2,,}2} has recursion level C(C(Ω22+C(Ω2++++C(C(Ω2++,σ),σ+),0),0),0)
  • {1{1{1,,`1`,,1,,2,,}2■2,,}1{1,,`1` `,,2,,}2} has recursion level C(C(Ω22+C(Ω2++++C(C(Ω2+++,σ),σ+),0),0),0)
  • {1{1{1,,`1`,,1,,2,,}2■2,,}1{1,,`2` `,,2,,}2} has recursion level C(C(Ω22+C(Ω2++++C(C(Ω2+++,σ)++),0),0),0)
  • {1,,`1` `,,3,,} has recursion level C(C(Ω22+C(Ω2++++C(C(Ω2+++,C(Ω2+++,σ)),σ+),0),0),0) (the growth boost appears again)
  • {1,,`1` `,,1,2,,} has recursion level C(C(Ω22+C(Ω2++++C(C(Ω2++++1,σ),σ+),0),0),0)
  • {1,,`1` `,,1{1{1,,`1` `,,2,,}2,,}2,,} has recursion level C(C(Ω22+C(Ω2++++C(C(Ω2++++C(C(Ω2+++,σ),σ+),σ),σ+),0),0),0)
  • {1,,`1` `,,1`,,2,,} has recursion level C(C(Ω22+C(Ω2++++C(C(Ω22,σ),σ+),0),0),0)
  • {1{1,,`1` `,,2,,}2{1,,`1` `,,1`,,2,,}2,,} has recursion level C(C(Ω22+C(Ω2++++C(C(Ω2+++,σ),C(C(Ω22,σ),σ+)),0),0),0)
  • {1{1,,`1` `,,1{1{1,,`1` `,,1`,,2,,}2,,}2,,}2{1,,`1` `,,1`,,2,,}2,,} has recursion level C(C(Ω22+C(Ω2++++C(C(Ω2++++C(C(Ω22,σ),σ+),σ),C(C(Ω22,σ),σ+)),0),0),0)
  • {1{1,,`1` `,,1`,,2,,}3,,} has recursion level C(C(Ω22+C(Ω2++++C(C(Ω22,σ),C(C(Ω22,σ),σ+)),0),0),0)
  • {1{1,,`1` `,,1`,,2,,}1,2,,} has recursion level C(C(Ω22+C(Ω2++++C(C(Ω22,σ)+1,σ+),0),0),0)
  • {1{1,,`1` `,,1`,,2,,}2,,`1` `,,1`,,2,,} has recursion level C(C(Ω22+C(Ω2++++C(C(Ω22,σ)2,σ+),0),0),0)
  • {1,,`2` `,,1`,,2,,} has recursion level C(C(Ω22+C(Ω2++++C(C(Ω22,σ)++),0),0),0)
  • {1,,`1` `,,1`,,3,,} has recursion level C(C(Ω22+C(Ω2++++C(C(Ω22,C(Ω22,σ)),σ+),0),0),0)
  • {1,,`1` `,,1`,,1,2,,} has recursion level C(C(Ω22+C(Ω2++++C(C(Ω22+1,0),σ+),0),0),0)
  • {1,,`1` `,,1`,,1,,2,,} has recursion level C(C(Ω22+C(Ω2+++++,0),0),0)
  • {1,,`1` `,,1`,,1`,,1,,2,,} has recursion level C(C(Ω22+C(Ω2+++++2,0),0),0)
  • {1,,`1` `,,1{2`,,}2,,} has recursion level C(C(Ω22+C(Ω2++++ω^(σ+++1),0),0),0)
  • {1,,`1` `,,1{1{1,,`1,2,,}2`,,}2,,} has recursion level C(C(Ω22+C(Ω2++++C(C(Ω2+1,σ),σ++),0),0),0)
  • {1,,`1` `,,1{1{1,,`1{1{1,,`1` `,,2,,}2,,}2,,}2`,,}2,,} has recursion level C(C(Ω22+C(Ω2++++C(C(Ω2+C(C(Ω2+++,σ),σ+),σ),σ++),0),0),0)
  • {1,,`1` `,,1{1{1,,`1{1{1,,`1` `,,1`,,1,,2,,}2,,}2,,}2`,,}2,,} has recursion level C(C(Ω22+C(Ω2++++C(C(Ω2+C(C(Ω2+++++,σ),σ+),σ),σ++),0),0),0)
  • {1,,`1` `,,1{1{1,,`1`,,2,,}2`,,}2,,} has recursion level C(C(Ω22+C(Ω2++++C(C(Ω2+C(C(Ω22,σ),σ+),σ),σ++),0),0),0)
  • {1{1,,`1` `,,1{1{1,,`1`,,2,,}2`,,}2,,}1,2,,} has recursion level C(C(Ω22+C(Ω2++++C(C(Ω2+C(C(Ω22,σ)+1,σ+),σ),σ++),0),0),0)
  • {1,,`2` `,,1{1{1,,`1`,,2,,}2`,,}2,,} has recursion level C(C(Ω22+C(Ω2++++C(C(Ω2+C(C(Ω22,σ)++),σ),σ++),0),0),0)
  • {1,,`1` `,,1{1{1,,`1`,,2,,}2`,,}1,2,,} has recursion level C(C(Ω22+C(Ω2++++C(C(Ω2+C(C(Ω22+1,0),σ+),σ),σ++),0),0),0)
  • {1,,`1` `,,1{1{1,,`1`,,2,,}2`,,}1,,2,,} has recursion level C(C(Ω22+C(Ω2++++C(C(Ω2++,σ),σ++),0),0),0)
  • {1{1,,`1` `,,1{1{1,,`1`,,2,,}2`,,}1,,2,,}1{1,,`1` `,,1{1{1,,`1`,,2,,}2`,,}2,,}2} has recursion level C(C(Ω22+C(Ω2++++C(C(Ω2++,σ),σ++)+C(C(Ω2++++C(C(Ω2++,σ),σ++),σ),σ+),0),0),0)
  • {1,,`1` `,,1{1{1,,`1`,,2,,}2`,,}1`,,2,,} has recursion level C(C(Ω22+C(Ω2++++C(C(Ω2++,σ),σ++)+C(C(Ω22,σ),σ+),0),0),0)
  • {1,,`1` `,,1{1{1,,`1`,,2,,}2`,,}1`,,1,,2,,} has recursion level C(C(Ω22+C(Ω2++++C(C(Ω2++,σ),σ++)+σ++,0),0),0)
  • {1,,`1` `,,1{2{1,,`1`,,2,,}2`,,}2,,} has recursion level C(C(Ω22+C(Ω2++++ω^(C(C(Ω2++,σ),σ++)+1),0),0),0)
  • {1,,`1` `,,1{1{1,,`1`,,2,,}1,2`,,}2,,} has recursion level C(C(Ω22+C(Ω2++++C(C(Ω2++,σ)+1,σ++),0),0),0)
  • {1,,`1` `,,1{1{1,,`2`,,2,,}2`,,}2,,} has recursion level C(C(Ω22+C(Ω2++++C(C(Ω2++,σ)+++),0),0),0)
  • {1,,`1` `,,1{1{1,,`1`,,1,2,,}2`,,}2,,} has recursion level C(C(Ω22+C(Ω2++++C(C(Ω2+++1,σ),σ++),0),0),0)
  • {1,,`1` `,,1{1{1,,`1{1{1,,`1`,,1,2,,}2`,,}2,,}2`,,}2,,} has recursion level C(C(Ω22+C(Ω2++++C(C(Ω2+C(C(Ω2+++1,σ),σ++),σ),σ++),0),0),0)
  • {1,,`1` `,,1{1{1,,`1` `,,2,,}2`,,}2,,} has recursion level C(C(Ω22+C(Ω2++++C(C(Ω2+++,σ),σ++),0),0),0)
  • {1,,`1` `,,1{1{1,,`1` `,,2,,}2`,,}1`,,1,,2,,} has recursion level C(C(Ω22+C(Ω2++++C(C(Ω2+++,σ),σ++)+σ++,0),0),0)
  • {1,,`1` `,,1{1{1,,`1` `,,2,,}1,2`,,}2,,} has recursion level C(C(Ω22+C(Ω2++++C(C(Ω2+++,σ)+1,σ++),0),0),0)
  • {1,,`1` `,,1{1{1,,`2` `,,2,,}2`,,}2,,} has recursion level C(C(Ω22+C(Ω2++++C(C(Ω2+++,σ)+++),0),0),0)
  • {1,,`1` `,,1{1{1,,`1` `,,1,2,,}2`,,}2,,} has recursion level C(C(Ω22+C(Ω2++++C(C(Ω2++++1,σ),σ++),0),0),0)
  • {1,,`1` `,,1{1{1,,`1` `,,1{1{1,,`1` `,,1,2,,}2`,,}2,,}2`,,}2,,} has recursion level C(C(Ω22+C(Ω2++++C(C(Ω2++++C(C(Ω2++++1,σ),σ++),σ),σ++),0),0),0)
  • {1,,`1` `,,1` `,,2,,} has recursion level C(C(Ω22+C(Ω2++++C(C(Ω22,σ),σ++),0),0),0)
  • {1{1,,`1` `,,1` `,,2,,}1{1,,`1` `,,1` `,,2,,}2} has recursion level C(C(Ω22+C(Ω2++++C(C(Ω22,σ),σ++)+C(C(Ω22,σ),σ+),0),0),0) (erratic thing again)

Despite of the erratic thing, which contributes little to the strength of TON (vs. sDAN), the C(Ω2,C(Ω2,C(Ω2,C(Ω22,0)))) approximately corresponds to {1,,`3,,} (or the ` `,,), then approximately

  • {1,,`1{2` `,,}2,,} has recursion level C(C(Ω22+C(Ω2+ω^(σ++++1),0),0),0)
  • {1,,`1{1` `,,2` `,,}2,,} has recursion level C(C(Ω22+C(Ω2+ω^(σ++++C(C(Ω22,σ),σ++)),0),0),0)
  • {1,,`1{1` `,,1,2` `,,}2,,} has recursion level C(C(Ω22+C(Ω2+ω^ω^(σ++++1),0),0),0)
  • {1,,`1{1{2` `,,}2` `,,}2,,} has recursion level C(C(Ω22+C(Ω2+ω^ω^ω^(σ++++1),0),0),0)
  • {1,,`1{1` ` `,,2` `,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(σ+++++++),0),0),0)
  • {1,,`1{1{1,,`1,2,,}2` `,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(C(Ω2+1,σ),σ+++),0),0),0)
  • {1,,`1{1{1,,`1,,2,,}2` `,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(C(Ω2+σ,σ),σ+++),0),0),0)
  • {1,,`1{1{1,,`1{2,,}2,,}2` `,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(C(Ω2+ω^(σ++1),σ),σ+++),0),0),0)
  • {1,,`1{1{1,,`1{2`,,}2,,}2` `,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(C(Ω2+ω^(σ+++1),σ),σ+++),0),0),0)
  • {1,,`1{1{1,,`1{2` `,,}2,,}2` `,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(C(Ω2+ω^(σ++++1),σ),σ+++),0),0),0)
  • {1,,`1` ` `,,2,,} has recursion level C(C(Ω22+C(Ω2+C(C(Ω22,σ),σ+++),0),0),0)

As shown above, the recursion level of {1,,`1`,,2,,} is C(C(Ω22+C(Ω2+C(C(Ω22,C(Ω22,0)),C(Ω2,C(Ω22,0))),0),0),0), the recursion level of {1,,`1` `,,2,,} is C(C(Ω22+C(Ω2+C(C(Ω22,C(Ω22,0)),C(Ω2,C(Ω2,C(Ω22,0)))),0),0),0), the recursion level of {1,,`1` ` `,,2,,} is C(C(Ω22+C(Ω2+C(C(Ω22,C(Ω22,0)),C(Ω2,C(Ω2,C(Ω2,C(Ω22,0))))),0),0),0), and so on. Generally, the recursion level of {1,,`1{1,,`k,,}2,,} (k > 1) is between C(C(Ω22+C(Ω2+C(Ω2,C(Ω2,…C(Ω2,C(Ω22,0))…)),0),0),0) (with k-1 Ω2′s in the blue part) and C(C(Ω22+C(Ω2+C(Ω2,C(Ω2,…C(Ω2,C(Ω22,0))…)),0),0),0) (with k Ω2′s in the blue part), so {1,,`1{1,,`1,2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(Ω2+1,C(Ω22,0)),0),0),0).

Now let ordinal σ = C(Ω22,0). And α+ is shorthand for C(Ω2,α).

  • {1{1,,`1{1,,`1,2,,}2,,}2} has recursion level C(C(Ω22+C(Ω2+C(Ω2+1,σ),0),0),0)
  • {1{1,,1,,1,2}2{1,,`1{1,,`1,2,,}2,,}2} has recursion level C(C(Ω22+C(Ω2+C(Ω2+1,σ),0)+C(Ω2++1,0),0),0)
  • {1{1{1,,`1`,,2,,}2}2{1,,`1{1,,`1,2,,}2,,}2} has recursion level C(C(Ω22+C(Ω2+C(Ω2+1,σ),0)+C(Ω2+C(C(Ω22,σ)+),0),0),0)
  • {1{1{1,,`1` `,,2,,}2}2{1,,`1{1,,`1,2,,}2,,}2} has recursion level C(C(Ω22+C(Ω2+C(Ω2+1,σ),0)+C(Ω2+C(C(Ω22,σ)++),0),0),0)
  • {1{1{1,,`1{1,,`1,2,,}2,,}2}2{1,,`1{1,,`1,2,,}2,,}2} has recursion level C(C(Ω22+C(Ω2+C(Ω2+1,σ),0)2,0),0)
  • {1,,1,,1,2{1,,`1{1,,`1,2,,}2,,}2} has recursion level C(C(Ω22+C(Ω2++1,C(Ω2+C(Ω2+1,σ),0)),0),0)
  • {1{1,,`1`,,2,,}2{1,,`1{1,,`1,2,,}2,,}2} has recursion level C(C(Ω22+C(Ω2+C(C(Ω22,σ)+),C(Ω2+C(Ω2+1,σ),0)),0),0)
  • {1{1,,`1` `,,2,,}2{1,,`1{1,,`1,2,,}2,,}2} has recursion level C(C(Ω22+C(Ω2+C(C(Ω22,σ)++),C(Ω2+C(Ω2+1,σ),0)),0),0)
  • {1{1,,`1{1,,`1,2,,}2,,}3} has recursion level C(C(Ω22+C(Ω2+C(Ω2+1,σ),C(Ω2+C(Ω2+1,σ),0)),0),0)
  • {1{1,,`1{1,,`1,2,,}2,,}1,2} has recursion level C(C(Ω22+C(Ω2+C(Ω2+1,σ)+1,0),0),0)
  • {1{1,,`1{1,,`1,2,,}2,,}1{1,,`1,2,,}2} has recursion level C(C(Ω22+C(Ω2+C(Ω2+1,σ)+C(C(Ω2+1,σ),σ+),0),0),0)
  • {1{1,,`1{1,,`1,2,,}2,,}1{1,,`1`,,2,,}2} has recursion level C(C(Ω22+C(Ω2+C(Ω2+1,σ)+C(C(Ω2++,σ),σ+),0),0),0)
  • {1{1,,`1{1,,`1,2,,}2,,}1{1,,`1` `,,2,,}2} has recursion level C(C(Ω22+C(Ω2+C(Ω2+1,σ)+C(C(Ω2+++,σ),σ+),0),0),0)
  • {1{1,,`1{1,,`1,2,,}2,,}1{1,,`1{1,,`1,2,,}2,,}2} has recursion level C(C(Ω22+C(Ω2+C(Ω2+1,σ)+C(C(Ω2+C(Ω2+1,σ),σ),σ+),0),0),0)
  • {1{1,,`1,2,,}1{1,,`1{1,,`1,2,,}2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(Ω2+1,σ)+C(C(Ω2+1,σ),C(C(Ω2+C(Ω2+1,σ),σ),σ+)),0),0),0)
  • {1{1,,`1{1,,`1,2,,}2,,}3,,} has recursion level C(C(Ω22+C(Ω2+C(Ω2+1,σ)+C(C(Ω2+C(Ω2+1,σ),σ),C(C(Ω2+C(Ω2+1,σ),σ),σ+)),0),0),0)
  • {1{1,,`1{1,,`1,2,,}2,,}1,2,,} has recursion level C(C(Ω22+C(Ω2+C(Ω2+1,σ)+C(C(Ω2+C(Ω2+1,σ),σ)+1,σ+),0),0),0)
  • {1,,`2{1,,`1,2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(Ω2+1,σ)+C(C(Ω2+C(Ω2+1,σ),σ)++),0),0),0)
  • {1,,`1{1,,`1,2,,}3,,} has recursion level C(C(Ω22+C(Ω2+C(Ω2+1,σ)+C(C(Ω2+C(Ω2+1,σ),C(Ω2+C(Ω2+1,σ),σ)),σ+),0),0),0)
  • {1,,`1{1,,`1,2,,}1,2,,} has recursion level C(C(Ω22+C(Ω2+C(Ω2+1,σ)+C(C(Ω2+C(Ω2+1,σ)+1,σ),σ+),0),0),0)
  • {1,,`1{1,,`1,2,,}1`,,2,,} has recursion level C(C(Ω22+C(Ω2+C(Ω2+1,σ)+C(C(Ω22,σ),σ+),0),0),0)
  • {1,,`2{1,,`1,2,,}1`,,2,,} has recursion level C(C(Ω22+C(Ω2+C(Ω2+1,σ)+C(C(Ω22,σ)++),0),0),0)
  • {1,,`1{1,,`1,2,,}1`,,1,2,,} has recursion level C(C(Ω22+C(Ω2+C(Ω2+1,σ)+C(C(Ω22+1,0),σ+),0),0),0)
  • {1,,`1{1,,`1,2,,}1`,,1,,2,,} has recursion level C(C(Ω22+C(Ω2+C(Ω2+1,σ)+σ++,0),0),0)
  • {1,,`1{1,,`1,2,,}1{2`,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(Ω2+1,σ)+ω^(σ+++1),0),0),0)
  • {1,,`1{1,,`1,2,,}1{1{1,,`1,2,,}2`,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(Ω2+1,σ)+C(C(Ω2+1,σ),σ++),0),0),0)
  • {1,,`1{1,,`1,2,,}1{1{1,,`1{1,,`1,2,,}2,,}2`,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(Ω2+1,σ)+C(C(Ω2+C(Ω2+1,σ),σ),σ++),0),0),0)
  • {1,,`1{1,,`1,2,,}1` `,,2,,} has recursion level C(C(Ω22+C(Ω2+C(Ω2+1,σ)+C(C(Ω22,σ),σ++),0),0),0)
  • {1,,`1{1,,`1,2,,}1{2` `,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(Ω2+1,σ)+σ+++,0),0),0)
  • {1,,`1{1,,`1,2,,}1` ` `,,2,,} has recursion level C(C(Ω22+C(Ω2+C(Ω2+1,σ)+C(C(Ω22,σ),σ+++),0),0),0)
  • {1,,`1{1,,`1,2,,}1{1,,`1,2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(Ω2+1,σ)2,0),0),0)
  • {1,,`1{1{1,,`2,2,,}2,,`1,2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(C(Ω2+1,σ)+,C(Ω2+1,σ)),0),0),0)
  • {1,,`1{1{1,,`1{1,,`1,2,,}2,,}2,,`1,2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(C(Ω2+C(Ω2+1,σ),σ),C(Ω2+1,σ)),0),0),0)
  • {1,,`1{1{1,,`1{1{1,,`1{1,,`1,2,,}2,,}2,,`1,2,,}2,,}2,,`1,2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(C(Ω2+C(C(Ω2+C(Ω2+1,σ),σ),C(Ω2+1,σ)),σ),C(Ω2+1,σ)),0),0),0)
  • {1,,`1{1,,`2,2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(C(Ω22,σ),C(Ω2+1,σ)),0),0),0)
  • {1,,`1{1,,`3,2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(C(Ω22,σ),C(Ω2+1,σ)+),0),0),0)
  • {1,,`1{1,,`1,3,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(Ω2+1,C(Ω2+1,σ)),0),0),0)
  • {1,,`1{1,,`1,1,2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(Ω2+2,σ),0),0),0)
  • {1,,`1{1,,`1{1{1,,`1{1,,`1,2,,}2,,}2}2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(Ω2+C(Ω2+C(Ω2+1,σ),0),σ),0),0),0)
  • {1,,`1{1,,`1,,2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(Ω2+σ,σ),0),0),0)

Due to the “growth boosts”, {1,,`k,,} approximately corresponds to C(Ω2,C(Ω2,…C(Ω2,C(Ω22,0))…)) (with k Ω2′s in the blue part), and {1,,`1,2,,} approximately corresponds to C(Ω2+1,C(Ω22,0)).

Beyond C(Ω2+C(Ω22,0),C(Ω22,0))

Now let separator ♦ = {1{1,,`1{1,,`1,,2,,}2,,}2} and let ordinal σ = C(Ω22,0). And α+ is shorthand for C(Ω2,α).

  • {1{1,,`1{1,,`1♦2,,}2,,}2} has recursion level C(C(Ω22+C(Ω2+C(Ω2+σ,σ),0),0),0)
  • {1{1,,1,,1,2}2{1,,`1{1,,`1♦2,,}2,,}2} has recursion level C(C(Ω22+C(Ω2+C(Ω2+σ,σ),0)+C(Ω2++1,0),0),0)
  • {1{1{1,,`1`,,2,,}2}2{1,,`1{1,,`1♦2,,}2,,}2} has recursion level C(C(Ω22+C(Ω2+C(Ω2+σ,σ),0)+C(Ω2+C(C(Ω22,σ),σ+),0),0),0)
  • {1{1{1,,`1{1,,`1,2,,}2,,}2}2{1,,`1{1,,`1♦2,,}2,,}2} has recursion level C(C(Ω22+C(Ω2+C(Ω2+σ,σ),0)+C(Ω2+C(Ω2+1,σ),0),0),0)
  • {1{1{1,,`1{1,,`1♦2,,}2,,}2}2{1,,`1{1,,`1♦2,,}2,,}2} has recursion level C(C(Ω22+C(Ω2+C(Ω2+σ,σ),0)2,0),0)
  • {1,,1,,1,2{1,,`1{1,,`1♦2,,}2,,}2} has recursion level C(C(Ω22+C(Ω2++1,C(Ω2+C(Ω2+σ,σ),0)),0),0)
  • {1{1,,`1{1,,`1,2,,}2,,}2{1,,`1{1,,`1♦2,,}2,,}2} has recursion level C(C(Ω22+C(Ω2+C(Ω2+1,σ),C(Ω2+C(Ω2+σ,σ),0)),0),0)
  • {1{1,,`1{1,,`1{1{1,,`1{1,,`1♦2,,}2,,}2}2,,}2,,}2{1,,`1{1,,`1♦2,,}2,,}2} has recursion level C(C(Ω22+C(Ω2+C(Ω2+C(Ω2+C(Ω2+σ,σ),0),σ),C(Ω2+C(Ω2+σ,σ),0)),0),0)
  • {1{1,,`1{1,,`1♦2,,}2,,}3} has recursion level C(C(Ω22+C(Ω2+C(Ω2+σ,σ),C(Ω2+C(Ω2+σ,σ),0)),0),0)
  • {1{1,,`1{1,,`1♦2,,}2,,}1,2} has recursion level C(C(Ω22+C(Ω2+C(Ω2+σ,σ)+1,0),0),0)
  • {1{1,,`1{1,,`1♦2,,}2,,}1{1,,`1{1,,`1,2,,}2,,}2} has recursion level C(C(Ω22+C(Ω2+C(Ω2+σ,σ)+C(C(Ω2+C(Ω2+1,σ),σ),σ+),0),0),0)
  • {1{1,,`1{1,,`1♦2,,}2,,}1{1,,`1{1,,`1♦2,,}2,,}2} has recursion level C(C(Ω22+C(Ω2+C(Ω2+σ,σ)+C(C(Ω2+C(Ω2+σ,σ),σ),σ+),0),0),0)
  • {1{1,,`1{1,,`1♦2,,}2,,}1,2,,} has recursion level C(C(Ω22+C(Ω2+C(Ω2+σ,σ)+C(C(Ω2+C(Ω2+σ,σ),σ)+1,σ+),0),0),0)
  • {1,,`2{1,,`1♦2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(Ω2+σ,σ)+C(C(Ω2+C(Ω2+σ,σ),σ)++),0),0),0)
  • {1,,`1{1,,`1♦2,,}1,2,,} has recursion level C(C(Ω22+C(Ω2+C(Ω2+σ,σ)+C(C(Ω2+C(Ω2+σ,σ)+1,σ),σ+),0),0),0)
  • {1,,`1{1,,`1♦2,,}1`,,2,,} has recursion level C(C(Ω22+C(Ω2+C(Ω2+σ,σ)+C(C(Ω22,σ),σ+),0),0),0)
  • {1,,`1{1,,`1♦2,,}1`,,1,,2,,} has recursion level C(C(Ω22+C(Ω2+C(Ω2+σ,σ)+σ++,0),0),0)
  • {1,,`1{1,,`1♦2,,}1` `,,2,,} has recursion level C(C(Ω22+C(Ω2+C(Ω2+σ,σ)+C(C(Ω22,σ),σ++),0),0),0)
  • {1,,`1{1,,`1♦2,,}1{1,,`1,2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(Ω2+σ,σ)+C(Ω2+1,σ),0),0),0)
  • {1,,`1{1,,`1♦2,,}1{1,,`1♦2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(Ω2+σ,σ)2,0),0),0)
  • {1,,`1{1{1,,`1{1,,`1,2,,}2,,}2,,`1♦2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(C(Ω2+C(Ω2+1,σ),σ),C(Ω2+σ,σ)),0),0),0)
  • {1,,`1{1{1,,`1{1,,`1♦2,,}2,,}2,,`1♦2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(C(Ω2+C(Ω2+σ,σ),σ),C(Ω2+σ,σ)),0),0),0)
  • {1,,`1{1,,`2♦2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(C(Ω22,σ),C(Ω2+σ,σ)),0),0),0)
  • {1,,`1{1,,`1,2♦2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(Ω2+1,C(Ω2+σ,σ)),0),0),0)
  • {1,,`1{1,,`1♦3,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(Ω2+σ,C(Ω2+σ,σ)),0),0),0)
  • {1,,`1{1,,`1♦1,2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(Ω2+σ+1,σ),0),0),0)
  • {1,,`1{1,,`1♦1♦2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(Ω2+σ2,σ),0),0),0)
  • {1,,`1{1,,`1{2{1,,`1{1,,`1,,2,,}2,,}2}2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(Ω2+ω^(σ+1),σ),0),0),0)
  • {1,,2{1,,`1{1,,`1,,2,,}2,,}2} has recursion level C(C(Ω22+C(Ω2+C(Ω2+C(σ+,σ),σ),0),0),0)
  • {1,,1,2{1,,`1{1,,`1,,2,,}2,,}2} has recursion level C(C(Ω22+C(Ω2+C(Ω2+C(C(Ω2+1,σ),σ),σ),0),0),0)
  • {1,,1{1,,1,2{1,,`1{1,,`1,,2,,}2,,}2}2{1,,`1{1,,`1,,2,,}2,,}2} has recursion level C(C(Ω22+C(Ω2+C(Ω2+C(C(Ω2+C(Ω2+1,σ),σ),σ),σ),0),0),0)
  • {1,,1{1,,1♦2{1,,`1{1,,`1,,2,,}2,,}2}2{1,,`1{1,,`1,,2,,}2,,}2} has recursion level C(C(Ω22+C(Ω2+C(Ω2+C(C(Ω2+C(Ω2+σ,σ),σ),σ),σ),0),0),0)
  • {1,,1{1,,1{1,,2{1,,`1{1,,`1,,2,,}2,,}2}2{1,,`1{1,,`1,,2,,}2,,}2}2{1,,`1{1,,`1,,2,,}2,,}2} has recursion level C(C(Ω22+C(Ω2+C(Ω2+C(C(Ω22,σ),σ),σ),0),0),0)
  • {1,,2{1,,1{1,,2{1,,`1{1,,`1,,2,,}2,,}2}2{1,,`1{1,,`1,,2,,}2,,}2}2{1,,`1{1,,`1,,2,,}2,,}2} has recursion level C(C(Ω22+C(Ω2+C(Ω2+C(C(Ω22,σ)+,σ),σ),0),0),0)
  • {1,,1{1,,1{1,,2{1,,`1{1,,`1,,2,,}2,,}2}2{1,,`1{1,,`1,,2,,}2,,}2}1,2{1,,`1{1,,`1,,2,,}2,,}2} has recursion level C(C(Ω22+C(Ω2+C(Ω2+C(C(Ω22+1,0),σ),σ),0),0),0)
  • {1,,1{1,,1{1,,2{1,,`1{1,,`1,,2,,}2,,}2}2{1,,`1{1,,`1,,2,,}2,,}2}1♦2{1,,`1{1,,`1,,2,,}2,,}2} has recursion level C(C(Ω22+C(Ω2+C(Ω2+,σ),0),0),0)
  • {1,,1,,1,2{1,,`1{1,,`1,,2,,}2,,}2} has recursion level C(C(Ω22+C(Ω2++1,C(Ω2+C(Ω2+,σ),0)),0),0)
  • {1{1,,`1{1,,`1,,2,,}2,,}1,2} has recursion level C(C(Ω22+C(Ω2+C(Ω2+,σ)+1,0),0),0)
  • {1{1,,`1{1,,`1,,2,,}2,,}1{1,,`1,2,,}2} has recursion level C(C(Ω22+C(Ω2+C(Ω2+,σ)+C(C(Ω2+1,σ),σ+),0),0),0)
  • {1{1,,`1{1,,`1,,2,,}2,,}1{1,,`1{1,,`1,,2,,}2,,}2} has recursion level C(C(Ω22+C(Ω2+C(Ω2+,σ)+C(C(Ω2+C(Ω2+σ,σ),σ),σ+),0),0),0)
  • {1{1,,`1{1,,`1,,2,,}2,,}1{1,,`1{1,,`1,,2,,}2,,}1,2} has recursion level C(C(Ω22+C(Ω2+C(Ω2+,σ)+C(C(Ω2+C(Ω2+,σ),σ),σ+)+1,0),0),0)
  • {1{1,,`1{1,,`1,,2,,}2,,}1,2,,} has recursion level C(C(Ω22+C(Ω2+C(Ω2+,σ)+C(C(Ω2+C(Ω2+,σ),σ)+1,σ+),0),0),0)
  • {1,,`2{1,,`1,,2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(Ω2+,σ)+C(C(Ω2+C(Ω2+,σ),σ)++),0),0),0)
  • {1,,`1{1,,`1,,2,,}1,2,,} has recursion level C(C(Ω22+C(Ω2+C(Ω2+,σ)+C(C(Ω2+C(Ω2+,σ)+1,σ),σ+),0),0),0)
  • {1,,`1{1,,`1,,2,,}1`,,2,,} has recursion level C(C(Ω22+C(Ω2+C(Ω2+,σ)+C(C(Ω22,σ),σ+),0),0),0)
  • {1,,`1{1,,`1,,2,,}1`,,1,,2,,} has recursion level C(C(Ω22+C(Ω2+C(Ω2+,σ)+σ++,0),0),0)
  • {1,,`1{1,,`1,,2,,}1{1,,`1,2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(Ω2+,σ)+C(Ω2+1,σ),0),0),0)
  • {1,,`1{1,,`1,,2,,}1{1,,`1,,2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(Ω2+,σ)+C(Ω2+σ,σ),0),0),0)
  • {1{1,,`1{1,,`1,,2,,}1{1,,`1,,2,,}2,,}1,2} has recursion level C(C(Ω22+C(Ω2+C(Ω2+,σ)2+1,0),0),0)
  • {1,,`1{2,,`1,,2,,}2,,} has recursion level C(C(Ω22+C(Ω2+ω^(C(Ω2+,σ)+1),0),0),0)
  • {1,,`1{1{1,,`2,,2,,}2,,`1,,2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(C(Ω2+,σ)+,C(Ω2+,σ)),0),0),0)
  • {1,,`1{1{1,,`1{1,,`1,,2,,}2,,}2,,`1,,2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(C(Ω2+C(Ω2+σ,σ),σ),C(Ω2+,σ)),0),0),0)
  • {1,,`1{1,,`2,,2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(C(Ω22,σ),C(Ω2+,σ)),0),0),0)
  • {1,,`1{1,,`1,2,,2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(Ω2+1,C(Ω2+,σ)),0),0),0)
  • {1,,`1{1,,`1,,3,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(Ω2+σ,C(Ω2+,σ)),0),0),0)
  • {1,,`1{1,,`1,,1,2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(Ω2++1,σ),0),0),0)
  • {1,,`1{1,,`1{2,,}2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(Ω2+ω^(σ++1),σ),0),0),0)
  • {1,,`1{1,,`1{1{1,,`1{1,,`1,,2,,}2,,}2,,}2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(Ω2+C(C(Ω2+C(Ω2+σ,σ),σ),σ+),σ),0),0),0)
  • {1,,`1{1,,`1{1{1,,`1{1,,`1{2,,}2,,}2,,}2,,}2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(Ω2+C(C(Ω2+C(Ω2+ω^(σ++1),σ),σ),σ+),σ),0),0),0)
  • {1,,`1{1,,`1`,,2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(Ω2+C(C(Ω22,σ),σ+),σ),0),0),0)
  • {1,,`1{1,,`1` `,,2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(Ω2+C(C(Ω22,σ),σ++),σ),0),0),0)
  • {1,,`1{1,,`1{1,,`1,2,,}2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(Ω2+C(Ω2+1,σ),σ),0),0),0)
  • {1,,`1{1,,`1{1,,`1{1,,`1,2,,}2,,}2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(Ω2+C(Ω2+C(Ω2+1,σ),σ),σ),0),0),0)
  • {1,,`1{1,,`1,,`2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(Ω22,σ),0),0),0)

So {1,,`1,,`2,,} has recursion level C(C(Ω22+C(Ω2+C(Ω22,C(Ω22,0)),0),0),0).

Up to {1,,`1{1,,`1,,`2,,}3,,}

Up to a C(Ω2,C(Ω22,C(Ω22,0)))

Now let ordinal σ = σ1 = C(Ω22,0), and σn+1 = C(Ω22,σn).

  • {1{1,,`1{1,,`1,,`2,,}2,,}2} has recursion level C(C(Ω22+C(Ω22,0),0),0)
  • {1{1,,1,,1,2}2{1,,`1{1,,`1,,`2,,}2,,}2} has recursion level C(C(Ω22+C(Ω22,0)+C(Ω2+C(Ω2,σ)+1,0),0),0)
  • {1{1{1,,`1`,,2,,}2}2{1,,`1{1,,`1,,`2,,}2,,}2} has recursion level C(C(Ω22+C(Ω22,0)+C(Ω2+C(σ2,C(Ω2,σ)),0),0),0)
  • {1{1{1,,`1{1,,`1,2,,}2,,}2}2{1,,`1{1,,`1,,`2,,}2,,}2} has recursion level C(C(Ω22+C(Ω22,0)+C(Ω2+C(Ω2+1,σ),0),0),0)
  • {1{1{1,,`1{1,,`1{1,,`1,2,,}2,,}2,,}2}2{1,,`1{1,,`1,,`2,,}2,,}2} has recursion level C(C(Ω22+C(Ω22,0)+C(Ω2+C(Ω2+C(Ω2+1,σ),σ),0),0),0)
  • {1{1{1,,`1{1,,`1,,`2,,}2,,}2}2{1,,`1{1,,`1,,`2,,}2,,}2} has recursion level C(C(Ω22+C(Ω22,0)2,0),0)
  • {1,,2{1,,`1{1,,`1,,`2,,}2,,}2} has recursion level C(C(Ω22+C(C(Ω2,C(Ω22,0)),C(Ω22,0)),0),0)
  • {1,,1,,1,2{1,,`1{1,,`1,,`2,,}2,,}2} has recursion level C(C(Ω22+C(Ω2+C(Ω2,σ)+1,C(Ω22,0)),0),0)
  • {1{1,,`1{1,,`1,2,,}2,,}2{1,,`1{1,,`1,,`2,,}2,,}2} has recursion level C(C(Ω22+C(Ω2+C(Ω2+1,σ),C(Ω22,0)),0),0)
  • {1{1,,`1{1,,`1,,`2,,}2,,}3} has recursion level C(C(Ω22+C(Ω22,C(Ω22,0)),0),0)
  • {1{1,,`1{1,,`1,,`2,,}2,,}1,2} has recursion level C(C(Ω22+C(Ω22+1,0),0),0)
  • {1{1,,`1{1,,`1,,`2,,}2,,}1,,2} has recursion level C(C(Ω22+C(Ω22+σ,0),0),0)
  • {1{1,,`1{1,,`1,,`2,,}2,,}1,,1,2} has recursion level C(C(Ω22+C(Ω22+C(Ω2,σ)+1,0),0),0)
  • {1{1,,`1{1,,`1,,`2,,}2,,}1{1`,,2,,}2} has recursion level C(C(Ω22+C(Ω22+C(C(Ω2,C(Ω2,σ)),C(Ω2,σ)),0),0),0)
  • {1{1,,`1{1,,`1,,`2,,}2,,}1{1,,`1,2,,}2} has recursion level C(C(Ω22+C(Ω22+C(C(Ω2+1,σ),C(Ω2,σ)),0),0),0)
  • {1{1,,`1{1,,`1,,`2,,}2,,}1{1,,`1`,,2,,}2} has recursion level C(C(Ω22+C(Ω22+C(C(Ω2+C(Ω2,C(Ω2,σ)),σ),C(Ω2,σ)),0),0),0)
  • {1{1,,`1{1,,`1,,`2,,}2,,}1{1,,`1{1,,`1,2,,}2,,}2} has recursion level C(C(Ω22+C(Ω22+C(C(Ω2+C(Ω2+1,σ),σ),C(Ω2,σ)),0),0),0)
  • {1{1,,`1{1,,`1,,`2,,}2,,}1{1,,`1{1,,`1,,`2,,}2,,}2} has recursion level C(C(Ω22+C(Ω22+C(σ2,C(Ω2,σ)),0),0),0) (from here on, the growth slows down. C(C(Ω22,σ),C(Ω2,σ)) is not valid)
  • {1{1,,`1,2,,}2{1,,`1{1,,`1,,`2,,}2,,}2,,} has recursion level C(C(Ω22+C(Ω22+C(C(Ω2+1,σ),C(σ2,C(Ω2,σ))),0),0),0)
  • {1{1,,`1{1,,`1,,`2,,}2,,}3,,} has recursion level C(C(Ω22+C(Ω22+C(σ2,C(σ2,C(Ω2,σ))),0),0),0)
  • {1{1,,`1{1,,`1,,`2,,}2,,}1,2,,} has recursion level C(C(Ω22+C(Ω22+C(σ2+1,C(Ω2,σ)),0),0),0)
  • {1{1,,`1{1,,`1,,`2,,}2,,}2,,`1{1,,`1,,`2,,}2,,} has recursion level C(C(Ω22+C(Ω22+C(σ22,C(Ω2,σ)),0),0),0)
  • {1,,`2{1,,`1,,`2,,}2,,} has recursion level C(C(Ω22+C(Ω22+C(C(Ω22),C(Ω2,σ)),0),0),0)
  • {1,,`1,2{1,,`1,,`2,,}2,,} has recursion level C(C(Ω22+C(Ω22+C(C(Ω2+1,σ2),C(Ω2,σ)),0),0),0)
  • {1,,`1{1,,`1,2,,}2{1,,`1,,`2,,}2,,} has recursion level C(C(Ω22+C(Ω22+C(C(Ω2+C(Ω2+1,σ),σ2),C(Ω2,σ)),0),0),0)
  • {1,,`1{1,,`1{1,,`1,,`2,,}2,,}2{1,,`1,,`2,,}2,,} has recursion level C(C(Ω22+C(Ω22+C(σ3,C(Ω2,σ)),0),0),0)
  • {1,,`1{1,,`1{1,,`1,,`2,,}2,,}3{1,,`1,,`2,,}2,,} has recursion level C(C(Ω22+C(Ω22+C(σ4,C(Ω2,σ)),0),0),0)
  • {1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,2{1,,`1,,`2,,}2,,} has recursion level C(C(Ω22+C(Ω22+C(C(Ω22+1,0),C(Ω2,σ)),0),0),0)
  • {1,,`1{1,,`1{1,,`1,,`2,,}2,,}1{1{1,,`1{1,,`1,,`2,,}2,,}1,2}2{1,,`1,,`2,,}2,,} has recursion level C(C(Ω22+C(Ω22+C(C(Ω22+C(Ω22+1,0),0),C(Ω2,σ)),0),0),0)
  • {1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,} has recursion level C(C(Ω22+C(Ω22+C(Ω2,C(Ω2,σ)),0),0),0)
  • {1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}1,2} has recursion level C(C(Ω22+C(Ω22+C(Ω2,C(Ω2,σ))+1,0),0),0)
  • {1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}1{1,,`1,2,,}2} has recursion level C(C(Ω22+C(Ω22+C(Ω2,C(Ω2,σ))+C(C(Ω2+1,σ),C(Ω2,σ)),0),0),0)
  • {1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}1{1,,`1{1,,`1,,`2,,}2,,}2} has recursion level C(C(Ω22+C(Ω22+C(Ω2,C(Ω2,σ))+C(σ2,C(Ω2,σ)),0),0),0)
  • {1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}2} has recursion level C(C(Ω22+C(Ω22+C(Ω2,C(Ω2,σ))2,0),0),0)
  • {1{1,,`1,2,,}2{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}2,,} has recursion level C(C(Ω22+C(Ω22+C(C(Ω2+1,σ),C(Ω2,C(Ω2,σ))),0),0),0)
  • {1{1,,`1{1,,`1,,`2,,}2,,}2{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}2,,} has recursion level C(C(Ω22+C(Ω22+C(σ2,C(Ω2,C(Ω2,σ))),0),0),0)
  • {1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,2{1,,`1,,`2,,}2,,}2{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}2,,} has recursion level C(C(Ω22+C(Ω22+C(C(Ω22+1,0),C(Ω2,C(Ω2,σ))),0),0),0)
  • {1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}3,,} has recursion level C(C(Ω22+C(Ω22+C(Ω2,C(Ω2,C(Ω2,σ))),0),0),0)
  • {1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}4,,} has recursion level C(C(Ω22+C(Ω22+C(Ω2,C(Ω2,C(Ω2,C(Ω2,σ)))),0),0),0)
  • {1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}1,2,,} has recursion level C(C(Ω22+C(Ω22+C(Ω2+1,σ),0),0),0)
  • {1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}1,,2,,} has recursion level C(C(Ω22+C(Ω22+C(Ω2+σ,σ),0),0),0)
  • {1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}1,,1,2,,} has recursion level C(C(Ω22+C(Ω22+C(Ω2+C(Ω2,σ)+1,σ),0),0),0)
  • {1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}1{1{1,,`1,2,,}2,,}2,,} has recursion level C(C(Ω22+C(Ω22+C(Ω2+C(C(Ω2+1,σ),C(Ω2,σ)),σ),0),0),0)
  • {1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}1{1{1,,`1{1,,`1,,`2,,}2,,}2,,}2,,} has recursion level C(C(Ω22+C(Ω22+C(Ω2+C(σ2,C(Ω2,σ)),σ),0),0),0)
  • {1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}1 {1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,2{1,,`1,,`2,,}2,,}2,,}2,,} has recursion level C(C(Ω22+C(Ω22+C(Ω2+C(C(Ω22+1,0),C(Ω2,σ)),σ),0),0),0)
  • {1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}1 {1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}2,,}2,,} has recursion level C(C(Ω22+C(Ω22+C(Ω2+C(Ω2,C(Ω2,σ)),σ),0),0),0)
  • {1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}1 {1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}1,2,,}2,,} has recursion level C(C(Ω22+C(Ω22+C(Ω2+C(Ω2+1,σ),σ),0),0),0)
  • {1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}1{1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}1 {1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}1,2,,}2,,}2,,} has recursion level C(C(Ω22+C(Ω22+C(Ω2+C(Ω2+C(Ω2+1,σ),σ),σ),0),0),0)
  • {1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}1`,,2,,} has recursion level C(C(Ω22+C(Ω22+C(Ω22,σ),0),0),0)
  • {1{1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}1`,,2,,}1,2} has recursion level C(C(Ω22+C(Ω22+C(Ω22,σ)+1,0),0),0)
  • {1{1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}1`,,2,,}1 {1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}2} has recursion level C(C(Ω22+C(Ω22+C(Ω22,σ)+C(Ω2,C(Ω2,σ)),0),0),0)
  • {1{1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}1`,,2,,}1 {1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}1`,,2,,}2} has recursion level C(C(Ω22+C(Ω22+C(Ω22,σ)2,0),0),0)
  • {1{2{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}1`,,2,,}2} has recursion level C(C(Ω22+C(Ω22+ω^(C(Ω22,σ)+1),0),0),0)
  • {1{1,,`1,2,,}2{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}1`,,2,,} has recursion level C(C(Ω22+C(Ω22+C(C(Ω2+1,C(Ω22,σ)),C(Ω22,σ)),0),0),0)
  • {1{1,,`1{1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}1`,,2,,}2,,}2 {1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}1`,,2,,} has recursion level C(C(Ω22+C(Ω22+C(C(Ω2+C(Ω22,σ),C(Ω22,σ)),C(Ω22,σ)),0),0),0)
  • {1{1,,`1`,,2,,}2{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}1`,,2,,} has recursion level C(C(Ω22+C(Ω22+C(C(Ω2+C(Ω2,C(Ω22,σ)),C(Ω22,σ)),C(Ω22,σ)),0),0),0)
  • {1{1,,`1{1,,`1,2,,}2,,}2{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}1`,,2,,} has recursion level C(C(Ω22+C(Ω22+C(C(Ω2+C(Ω2+1,C(Ω22,σ)),C(Ω22,σ)),C(Ω22,σ)),0),0),0)
  • {1{1,,`1{1,,`1,,`2,,}2,,}2{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}1`,,2,,} has recursion level C(C(Ω22+C(Ω22+C(σ2,C(Ω22,σ)),0),0),0)
  • {1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,2{1,,`1,,`2,,}2,,}2{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}1`,,2,,} has recursion level C(C(Ω22+C(Ω22+C(C(Ω22+1,0),C(Ω22,σ)),0),0),0)
  • {1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}2`,,2,,} has recursion level C(C(Ω22+C(Ω22+C(Ω2,C(Ω22,σ)),0),0),0)
  • {1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}1 {1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}1`,,2,,}2`,,2,,} has recursion level C(C(Ω22+C(Ω22+C(Ω2+C(Ω22,σ),C(Ω22,σ)),0),0),0)
  • {1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}1`,,3,,} has recursion level C(C(Ω22+C(Ω22+C(Ω22,C(Ω22,σ)),0),0),0)
  • {1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}1`,,1,2,,} has recursion level C(C(Ω22+C(Ω22+C(Ω22+1,σ),0),0),0)
  • {1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}1`,,1 {1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}1`,,2,,}2,,} has recursion level C(C(Ω22+C(Ω22+C(Ω22+C(Ω22,σ),σ),0),0),0)
  • {1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}1`,,1`,,2,,} has recursion level C(C(Ω22+C(Ω222,0),0),0)
  • {1{1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}1`,,1`,,2,,}1 {1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}2} has recursion level C(C(Ω22+C(Ω222+C(Ω2,C(Ω2,σ)),0),0),0)
  • {1{1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}1`,,1`,,2,,}1 {1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}1`,,2,,}2} has recursion level C(C(Ω22+C(Ω222+C(Ω22,σ),0),0),0)
  • {1{1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}1`,,1`,,2,,}1 {1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}1`,,1`,,2,,}2} has recursion level C(C(Ω22+C(Ω222+C(Ω222,σ),0),0),0)
  • {1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}2`,,1`,,2,,} has recursion level C(C(Ω22+C(Ω222+C(Ω2,C(Ω222,σ)),0),0),0)
  • {1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}1`,,2`,,2,,} has recursion level C(C(Ω22+C(Ω222+C(Ω22,C(Ω222,σ)),0),0),0)
  • {1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}1`,,1`,,3,,} has recursion level C(C(Ω22+C(Ω222+C(Ω222,C(Ω222,σ)),0),0),0)
  • {1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}1`,,1`,,1,2,,} has recursion level C(C(Ω22+C(Ω222+C(Ω222+1,σ),0),0),0)
  • {1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}1`,,1`,,1`,,2,,} has recursion level C(C(Ω22+C(Ω223,0),0),0)
  • {1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}1{2`,,}2,,} has recursion level C(C(Ω22+C(Ω2+ω^(σ2+1),0),0),0)
  • {1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}1{1`,,2`,,}2,,} has recursion level C(C(Ω22+C(Ω2+ω^(σ22),0),0),0)
  • {1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}1` `,,2,,} has recursion level C(C(Ω22+C(Ω2+C(C(Ω22),σ2),0),0),0)
  • {1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}1` `,,1` `,,2,,} has recursion level C(C(Ω22+C(Ω2+C(C(Ω22),σ2)2,0),0),0)
  • {1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}1{2` `,,2`,,}2,,} has recursion level C(C(Ω22+C(Ω2+ω^(C(C(Ω22),σ2)+1),0),0),0)
  • {1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}1{1` `,,3`,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(C(Ω22),C(C(Ω22),σ2)),0),0),0)
  • {1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}1{1` `,,1` `,,2`,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(C(Ω22)2,σ2),0),0),0)
  • {1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}1` ` `,,2,,} has recursion level C(C(Ω22+C(Ω2+C(C(Ω2,C(Ω22)),σ2),0),0),0)
  • {1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}1{1,,`1,2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(C(Ω2+1,σ2),σ2),0),0),0)
  • {1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}1{1,,`1,,1,2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(C(Ω2+C(Ω2,σ)+1,σ2),σ2),0),0),0)
  • {1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}1 {1,,`1{1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}2,,}2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(C(Ω2+C(Ω2,C(Ω2,σ)),σ2),σ2),0),0),0)
  • {1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}1 {1,,`1{1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}1`,,2,,}2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(C(Ω2+C(Ω22,σ),σ2),σ2),0),0),0)
  • {1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}1 {1,,`1{1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}1` `,,2,,}2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(C(Ω2+C(Ω2+C(C(Ω22),σ2),σ),σ2),σ2),0),0),0)
  • {1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}1{1,,`1`,,2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(C(Ω222),σ2),0),0),0)
  • {1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}1{1,,`1{1` `,,2`,,}2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(C(Ω2+C(C(Ω22),σ2),σ2),σ2),0),0),0)
  • {1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}1{1,,`1{1{1,,`1`,,2,,}2`,,}2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(C(Ω2+C(C(Ω222),σ2),σ2),σ2),0),0),0)
  • {1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}1{1,,`1` `,,2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(σ32),0),0),0)
  • {1{1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}1{1,,`1` `,,2,,}2,,}1 {1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}1{1,,`1` `,,2,,}2,,}2} has recursion level C(C(Ω22+C(Ω2+C(σ32)+C(Ω2+C(σ32),σ),0),0),0)
  • {1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}1{1,,`1` `,,2,,}1,2,,} has recursion level C(C(Ω22+C(Ω2+C(σ32)+C(Ω2+C(σ32)+1,σ),0),0),0)
  • {1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}1{1,,`1` `,,2,,}1`,,2,,} has recursion level C(C(Ω22+C(Ω2+C(σ32)+σ2,0),0),0)
  • {1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}1{1,,`1` `,,2,,}1{1,,`1` `,,2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(σ32)2,0),0),0)
  • {1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}1{1` `,,2{1,,`1` `,,2,,}2`,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(C(Ω22),C(σ32)),0),0),0)
  • {1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}1{1{1,,`1`,,2,,}2{1,,`1` `,,2,,}2`,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(C(Ω222),C(σ32)),0),0),0)
  • {1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}1{1{1,,`1` `,,2,,}3`,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(σ3,C(σ32)),0),0),0)
  • {1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}1{1{1,,`1` `,,2,,}1,2`,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(σ3+1,σ2),0),0),0)
  • {1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}1{1,,`2` `,,2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(C(Ω23),σ2),0),0),0)
  • {1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}1{1,,`1` `,,3,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(σ42),0),0),0)
  • {1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}1{1,,`1` `,,1,2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(C(Ω22+1,0),σ2),0),0),0)
  • {1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}1{1,,`1` `,,1,,2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(Ω22),0),0),0)

Here’re some approximations, to make the comparisons above more simple:

  • The double comma approximately corresponds to C(Ω2,C(Ω22,0))
  • {1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}2,,} approximately corresponds to C(Ω2,C(Ω2,C(Ω22,0)))
  • {1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}3,,} approximately corresponds to C(Ω2,C(Ω2,C(Ω2,C(Ω22,0))))
  • {1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}1,2,,} approximately corresponds to C(Ω2+1,C(Ω22,0))
  • {1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}1,,2,,} approximately corresponds to C(Ω2+C(Ω2,C(Ω22,0)),C(Ω22,0))
  • {1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}1 {1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}1,,2,,}2,,} approximately corresponds to C(Ω2+C(Ω2+C(Ω2,C(Ω22,0)),C(Ω22,0)),C(Ω22,0))
  • {1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}1`,,2,,} approximately corresponds to C(Ω2+C(Ω22,C(Ω22,0)),C(Ω22,0))
  • {1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}1` `,,2,,} approximately corresponds to C(Ω2+C(C(Ω2,C(Ω22,C(Ω22,0))),C(Ω22,C(Ω22,0))),C(Ω22,0))
  • {1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}1{1,,`1`,,2,,}2,,} approximately corresponds to C(Ω2+C(C(Ω2+C(Ω22,C(Ω22,0)),C(Ω22,C(Ω22,0))),C(Ω22,C(Ω22,0))),C(Ω22,0))
  • {1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}1{1,,`1` `,,2,,}2,,} approximately corresponds to C(Ω2+C(C(Ω22,C(Ω22,C(Ω22,0))),C(Ω22,C(Ω22,0))),C(Ω22,0))
  • {1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}1{1,,`1` `,,3,,}2,,} approximately corresponds to C(Ω2+C(C(Ω22,C(Ω22,C(Ω22,C(Ω22,0)))),C(Ω22,C(Ω22,0))),C(Ω22,0))
  • {1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}1{1,,`1` `,,1,2,,}2,,} approximately corresponds to C(Ω2+C(C(Ω22+1,0),C(Ω22,C(Ω22,0))),C(Ω22,0))

Up to a C(Ω22,C(Ω22,C(Ω22,0)))

Now let separator ♦ = {1{1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}1{1,,`1` `,,1,,2,,}2,,}2} and let ordinal σ = σ1 = C(Ω22,0), and σn+1 = C(Ω22,σn).

  • {1{1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}1{1,,`1` `,,1♦2,,}2,,}2} has recursion level C(C(Ω22+C(Ω2+C(Ω22),0),0),0)
  • {1{1,,1,,1,2}2{1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}1{1,,`1` `,,1♦2,,}2,,}2} has recursion level C(C(Ω22+C(Ω2+C(Ω22),0)+C(Ω2+C(Ω2,σ)+1,0),0),0)
  • {1{1{1,,`1{1,,`1,,`2,,}2,,}2}2{1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}1{1,,`1` `,,1♦2,,}2,,}2} has recursion level C(C(Ω22+C(Ω2+C(Ω22),0)+C(Ω22,0),0),0)
  • {1{1{1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}1{1,,`1` `,,1,2,,}2,,}2}2 {1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}1{1,,`1` `,,1♦2,,}2,,}2} has recursion level C(C(Ω22+C(Ω2+C(Ω22),0)+C(Ω2+C(C(Ω22+1,0),σ2),0),0),0)
  • {1{1{1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}1{1,,`1` `,,1♦2,,}2,,}2}2 {1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}1{1,,`1` `,,1♦2,,}2,,}2} has recursion level C(C(Ω22+C(Ω2+C(Ω22),0)+C(Ω2+C(C(Ω22+C(Ω2+C(Ω22),0),0),σ2),0),0),0)
  • {1,,1,,1,2{1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}1{1,,`1` `,,1♦2,,}2,,}2} has recursion level C(C(Ω22+C(Ω2+C(Ω22),0)+C(Ω2+C(C(Ω22+C(Ω2+C(Ω2,σ)+1,C(Ω2+C(C(Ω22+C(Ω2+C(Ω22),0),0),σ2),0)),0),σ),C(Ω2+C(C(Ω22+C(Ω2+C(Ω22),0),0),σ2),0)),0),0)
  • {1{1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}1{1,,`1` `,,1♦2,,}2,,}1,2} has recursion level C(C(Ω22+C(Ω2+C(Ω22),0)+C(Ω2+C(C(Ω22+C(Ω2+C(Ω22),0),0),σ2)+1,0),0),0)
  • {1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}1{1,,`1` `,,1♦1,2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(Ω22),0)+C(Ω2+C(C(Ω22+C(Ω2+C(Ω22),0)+1,0),σ2),0),0),0)
  • {1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}1{1,,`1` `,,1♦1♦2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(Ω22),0)2,0),0)
  • {1,,1,,1,2{1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}1{1,,`1` `,,1,,2,,}2,,}2} has recursion level C(C(Ω22+C(Ω2+C(Ω2,σ)+1,C(Ω2+C(Ω22),0)),0),0)
  • {1{1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}1{1,,`1` `,,1,2,,}2,,}2 {1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}1{1,,`1` `,,1,,2,,}2,,}2} has recursion level C(C(Ω22+C(Ω2+C(C(Ω22+1,0),σ2),C(Ω2+C(Ω22),0)),0),0)
  • {1{1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}1{1,,`1` `,,1,,2,,}2,,}3} has recursion level C(C(Ω22+C(Ω2+C(Ω22),C(Ω2+C(Ω22),0)),0),0)
  • {1{1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}1{1,,`1` `,,1,,2,,}2,,}1,2} has recursion level C(C(Ω22+C(Ω2+C(Ω22)+1,0),0),0)
  • {1{1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}1{1,,`1` `,,1,,2,,}2,,}1{1,,`1{1,,`1,,`2,,}2,,}2} has recursion level C(C(Ω22+C(Ω2+C(Ω22)+C(σ2,C(Ω2,σ)),0),0),0)
  • {1{1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}1{1,,`1` `,,1,,2,,}2,,}1 {1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}2} has recursion level C(C(Ω22+C(Ω2+C(Ω22)+C(Ω2,C(Ω2,σ)),0),0),0)
  • {1{1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}1{1,,`1` `,,1,,2,,}2,,}1 {1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}1,2,,}2} has recursion level C(C(Ω22+C(Ω2+C(Ω22)+C(Ω2+1,σ),0),0),0)
  • {1{1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}1{1,,`1` `,,1,,2,,}2,,}1 {1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}1`,,2,,}2} has recursion level C(C(Ω22+C(Ω2+C(Ω22)+C(Ω22,σ),0),0),0)
  • {1{1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}1{1,,`1` `,,1,,2,,}2,,}1 {1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}1{1,,`1,2,,}2,,}2} has recursion level C(C(Ω22+C(Ω2+C(Ω22)+C(Ω2+C(C(Ω2+1,σ2),σ2),σ),0),0),0)
  • {1{1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}1{1,,`1` `,,1,,2,,}2,,}1 {1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}1{1,,`1`,,2,,}2,,}2} has recursion level C(C(Ω22+C(Ω2+C(Ω22)+C(Ω2+C(C(Ω222),σ2),σ),0),0),0)
  • {1{1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}1{1,,`1` `,,1,,2,,}2,,}1 {1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}1{1,,`1` `,,2,,}2,,}2} has recursion level C(C(Ω22+C(Ω2+C(Ω22)+C(Ω2+C(σ32),σ),0),0),0)
  • {1{1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}1{1,,`1` `,,1,,2,,}2,,}1 {1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}1{1,,`1` `,,1,2,,}2,,}2} has recursion level C(C(Ω22+C(Ω2+C(Ω22)+C(Ω2+C(C(Ω22+1,0),σ2),σ),0),0),0)
  • {1{1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}1{1,,`1` `,,1,,2,,}2,,}1 {1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}1{1,,`1` `,,1,,2,,}2,,}2} has recursion level C(C(Ω22+C(Ω2+C(Ω22)+C(Ω2+C(Ω22),σ),0),0),0)
  • {1{1,,`1{1,,`1,,`2,,}2,,}2{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}1{1,,`1` `,,1,,2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(Ω22)+C(σ2,C(Ω2+C(Ω22),σ)),0),0),0)
  • {1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}2{1,,`1` `,,1,,2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(Ω22)+C(Ω2,C(Ω2+C(Ω22),σ)),0),0),0)
  • {1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}1`,,2{1,,`1` `,,1,,2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(Ω22)+C(Ω22,C(Ω2+C(Ω22),σ)),0),0),0)
  • {1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}1{1,,`1` `,,2,,}2{1,,`1` `,,1,,2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(Ω22)+C(Ω2+C(σ32),C(Ω2+C(Ω22),σ)),0),0),0)
  • {1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}1{1,,`1` `,,1,,2,,}3,,} has recursion level C(C(Ω22+C(Ω2+C(Ω22)+C(Ω2+C(Ω22),C(Ω2+C(Ω22),σ)),0),0),0)
  • {1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}1{1,,`1` `,,1,,2,,}1,2,,} has recursion level C(C(Ω22+C(Ω2+C(Ω22)+C(Ω2+C(Ω22)+1,σ),0),0),0)
  • {1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}1{1,,`1` `,,1,,2,,}1`,,2,,} has recursion level C(C(Ω22+C(Ω2+C(Ω22)+σ2,0),0),0)
  • {1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}1{1,,`1` `,,1,,2,,}1{1,,`1`,,2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(Ω22)+C(C(Ω222),σ2),0),0),0)
  • {1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}1{1,,`1` `,,1,,2,,}1{1,,`1` `,,2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(Ω22)+C(C(Ω2+C(Ω22),σ2),σ2),0),0),0)
  • {1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}1{1,,`1` `,,1,,2,,}1{1,,`1` `,,1,,2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(Ω22)+C(C(Ω2+C(Ω22)+σ,σ2),σ2),0),0),0)
  • {1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}1{1,,`1` `,,1`,,2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(Ω22)+C(C(Ω2+C(Ω22)+σ22),σ2),0),0),0)
  • {1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}1{1,,`1` `,,1{1{1,,`1` `,,2,,}2`,,}2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(Ω22)+C(C(Ω2+C(Ω22)+C(C(Ω2+C(Ω22),σ2),σ2),σ2),σ2),0),0),0)
  • {1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}1{1,,`1` `,,1` `,,2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(Ω22)+C(σ32),0),0),0)
  • {1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}1{1,,`1` `,,1` `,,1,2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(Ω22)+C(C(Ω22+1,0),σ2),0),0),0)
  • {1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}1{1,,`1` `,,1` `,,1,,2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(Ω22)2,0),0),0)
  • {1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}1{1,,`1` `,,1` `,,1` `,,1,,2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(Ω22)3,0),0),0)
  • {1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}1{1,,`1{2` `,,}2,,}2,,} has recursion level C(C(Ω22+C(Ω2+ω^(C(Ω22)+1),0),0),0)
  • {1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}1{1,,`1{1{1,,`1,2,,}2` `,,}2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(C(Ω2+1,σ2),C(Ω22)),0),0),0)
  • {1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}1{1,,`1{1{1,,`1`,,2,,}2` `,,}2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(C(Ω222),C(Ω22)),0),0),0)
  • {1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}1{1,,`1{1{1,,`1` `,,2,,}2` `,,}2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(C(Ω2+C(σ32),σ2),C(Ω22)),0),0),0)
  • {1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}1{1,,`1{1{1,,`1` `,,1,,2,,}2` `,,}2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(C(Ω2+C(Ω22),σ2),C(Ω22)),0),0),0)
  • {1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}1{1,,`1{1{1,,`1{1{1,,`1` `,,1,,2,,}2` `,,}2,,}2` `,,}2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(C(Ω2+C(C(Ω2+C(Ω22),σ2),C(Ω22)),σ2),C(Ω22)),0),0),0)
  • {1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}1{1,,`1` ` `,,2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(σ3,C(Ω22)),0),0),0)
  • {1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}1{1,,`1` ` ` `,,2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(σ3,C(Ω2,C(Ω22))),0),0),0)
  • {1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}1{1,,`1{1,,`1,2,,}2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(Ω2+1,σ2),0),0),0)
  • {1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}1{1,,`1{1,,`1{1{1,,`1{1,,`1,,`2,,}2,,}2,,}2,,}2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(Ω2+C(Ω22,σ),σ2),0),0),0)
  • {1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}1 {1,,`1{1,,`1{1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}1{1,,`1{1,,`1,2,,}2,,}2,,}2,,}2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(Ω2+C(Ω2+C(Ω2+1,σ2),σ),σ2),0),0),0)
  • {1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}1{1,,`1{1,,`1`,,2,,}2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(Ω222),0),0),0)
  • {1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}1{1,,`1{1,,`1{1{1,,`1,2,,}2`,,}2,,}2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(Ω2+C(C(Ω2+1,σ2),σ2),σ2),0),0),0)
  • {1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}1{1,,`1{1,,`1{1{1,,`1`,,2,,}2`,,}2,,}2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(Ω2+C(C(Ω222),σ2),σ2),0),0),0)
  • {1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}1{1,,`1{1,,`1{1{1,,`1{1,,`1,2,,}2,,}2`,,}2,,}2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(Ω2+C(C(Ω2+C(Ω2+1,σ2),σ2),σ2),σ2),0),0),0)
  • {1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}1{1,,`1{1,,`1{1{1,,`1{1,,`1`,,2,,}2,,}2`,,}2,,}2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(Ω2+C(C(Ω2+C(Ω222),σ2),σ2),σ2),0),0),0)
  • {1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}1{1,,`1{1,,`1` `,,2,,}2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(Ω2+C(σ32),σ2),0),0),0)
  • {1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}1{1,,`1{1,,`1` ` `,,2,,}2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(Ω2+C(σ3,C(Ω22)),σ2),0),0),0)
  • {1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}1{1,,`1{1,,`1{1,,`1,2,,}2,,}2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(Ω2+C(Ω2+1,σ2),σ2),0),0),0)
  • {1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}1{1,,`1{1,,`1{1,,`1{1,,`1,2,,}2,,}2,,}2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(Ω2+C(Ω2+C(Ω2+1,σ2),σ2),σ2),0),0),0)
  • {1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}1{1,,`1{1,,`1,,`2,,}2,,}2,,} has recursion level C(C(Ω22+C(Ω23,0),0),0)

Here’re some approximations, to make the comparisons above more simple:

  • {1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}1{1,,`1` `,,1,,2,,}2,,} approximately corresponds to C(Ω2+C(Ω2,C(Ω22,C(Ω22,0))),C(Ω22,0))
  • {1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}1{1,,`1` `,,1` `,,1,,2,,}2,,} approximately corresponds to C(Ω2+C(Ω2,C(Ω22,C(Ω22,0)))2,C(Ω22,0))
  • {1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}1{1,,`1` ` `,,2,,}2,,} approximately corresponds to C(Ω2+C(C(Ω22,C(Ω22,C(Ω22,0))),C(Ω2,C(Ω22,C(Ω22,0)))),C(Ω22,0))
  • {1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}1{1,,`1` ` ` `,,2,,}2,,} approximately corresponds to C(Ω2+C(C(Ω22,C(Ω22,C(Ω22,0))),C(Ω2,C(Ω2,C(Ω22,C(Ω22,0))))),C(Ω22,0))
  • {1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}1{1,,`1{1,,`1,2,,}2,,}2,,} approximately corresponds to C(Ω2+C(Ω2+1,C(Ω22,C(Ω22,0))),C(Ω22,0))
  • {1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}1{1,,`1{1,,`1`,,2,,}2,,}2,,} approximately corresponds to C(Ω2+C(Ω2+C(Ω22,C(Ω22,0)),C(Ω22,C(Ω22,0))),C(Ω22,0))
  • {1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}1{1,,`1{1,,`1{1,,`1,2,,}2,,}2,,}2,,} approximately corresponds to C(Ω2+C(Ω2+C(Ω2+1,C(Ω22,C(Ω22,0))),C(Ω22,C(Ω22,0))),C(Ω22,0))

The C(Ω22,C(Ω22,0)) seems to correspond to the `,, (i.e. {1,,`2,,}) in {1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}1{ ____ `,,}2,,}. And C(Ω2,C(Ω22,C(Ω22,0))) seems to correspond to {1,,`3,,}, C(Ω2,C(Ω2,C(Ω22,C(Ω22,0)))) seems to correspond to {1,,`4,,}, C(Ω2+1,C(Ω22,C(Ω22,0))) seems to correspond to {1,,`1,2,,}, and C(Ω2+C(Ω22,C(Ω22,0)),C(Ω22,C(Ω22,0))) seems to correspond to {1,,`1`,,2,,}. But those will be weaken in next comparisons.

Up to a C(Ω22+C(Ω22,0),0)

Now let ordinal σ = σ1 = C(Ω22,0), and σn+1 = C(Ω22,σn).

  • {1{1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}1{1,,`1{1,,`1,,`2,,}2,,}2,,}1,2} has recursion level C(C(Ω22+C(Ω23+1,0),0),0)
  • {1{1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}1{1,,`1{1,,`1,,`2,,}2,,}2,,}1 {1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}1`,,2,,}2} has recursion level C(C(Ω22+C(Ω23+C(Ω22,σ),0),0),0)
  • {1{1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}1{1,,`1{1,,`1,,`2,,}2,,}2,,}1 {1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}1{1,,`1{1,,`1,,`2,,}2,,}2,,}2} has recursion level C(C(Ω22+C(Ω23+C(Ω23,σ),0),0),0)
  • {1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}2{1,,`1{1,,`1,,`2,,}2,,}2,,} has recursion level C(C(Ω22+C(Ω23+C(Ω2,C(Ω23,σ)),0),0),0)
  • {1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}1{1,,`1{1,,`1,,`2,,}2,,}3,,} has recursion level C(C(Ω22+C(Ω23+C(Ω23,C(Ω23,σ)),0),0),0)
  • {1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}1{1,,`1{1,,`1,,`2,,}2,,}1,2,,} has recursion level C(C(Ω22+C(Ω23+C(Ω23+1,σ),0),0),0)
  • {1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}1{1,,`1{1,,`1,,`2,,}2,,}1`,,2,,} has recursion level C(C(Ω22+C(Ω232,0),0),0)
  • {1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}1{1,,`1{1,,`1,,`2,,}2,,}1{1,,`1,2,,}2,,} has recursion level C(C(Ω22+C(Ω23+C(C(Ω2+1,σ2),σ2),0),0),0)
  • {1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}1{1,,`1{1,,`1,,`2,,}2,,}1{1,,`1{1,,`1,2,,}2,,}2,,} has recursion level C(C(Ω22+C(Ω23+C(C(Ω2+C(Ω2+1,σ2),σ2),σ2),0),0),0)
  • {1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}1{1,,`1{1,,`1,,`2,,}2,,}1{1,,`1{1,,`1,,`2,,}2,,}2,,} has recursion level C(C(Ω22+C(Ω23+C(σ32),0),0),0)
  • {1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}1 {1,,`1{1,,`1,,`2,,}2,,}1{1,,`1{1,,`1,,`2,,}2,,}1{1,,`1{1,,`1,,`2,,}2,,}2,,} has recursion level C(C(Ω22+C(Ω23+C(σ32)2,0),0),0)
  • {1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}1{2{1,,`1{1,,`1,,`2,,}2,,}2`,,}2,,} has recursion level C(C(Ω22+C(Ω23+ω^(C(σ32)+1),0),0),0)
  • {1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}1{1` `,,2{1,,`1{1,,`1,,`2,,}2,,}2`,,}2,,} has recursion level C(C(Ω22+C(Ω23+C(C(Ω22),C(σ32)),0),0),0)
  • {1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}1{1{1,,`1{1,,`1,,`2,,}2,,}3`,,}2,,} has recursion level C(C(Ω22+C(Ω23+C(σ3,C(σ32)),0),0),0)
  • {1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}1{1{1,,`1{1,,`1,,`2,,}2,,}1,2`,,}2,,} has recursion level C(C(Ω22+C(Ω23+C(σ3+1,σ2),0),0),0)
  • {1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}1{1{1,,`1{1,,`1,,`2,,}2,,}2,,`1{1,,`1,,`2,,}2,,}2,,} has recursion level C(C(Ω22+C(Ω23+C(σ32,σ2),0),0),0)
  • {1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}1{1,,`2{1,,`1,,`2,,}2,,}2,,} has recursion level C(C(Ω22+C(Ω23+C(C(Ω23),σ2),0),0),0)
  • {1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}1{1,,`1{1,,`1,2,,}2{1,,`1,,`2,,}2,,}2,,} has recursion level C(C(Ω22+C(Ω23+C(C(Ω2+C(Ω2+1,σ2),σ3),σ2),0),0),0)
  • {1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}2{1,,`1,,`2,,}2,,}2,,} has recursion level C(C(Ω22+C(Ω23+C(σ42),0),0),0)
  • {1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,2{1,,`1,,`2,,}2,,}2,,} has recursion level C(C(Ω22+C(Ω23+C(C(Ω22+1,0),σ2),0),0),0)
  • {1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}2,,} has recursion level C(C(Ω22+C(Ω23+C(Ω22),0),0),0)
  • {1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}1`,,2,,} has recursion level C(C(Ω22+C(Ω23+C(Ω22)+σ2,0),0),0)
  • {1{2{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}2`,,}2,,} has recursion level C(C(Ω22+C(Ω23+ω^(C(Ω22)+1),0),0),0)
  • {1`,,2,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,} has recursion level C(C(Ω22+C(Ω23+ω^(C(Ω22)+σ2),0),0),0)
  • {1{1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}2`,,}2 {1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}2`,,} has recursion level C(C(Ω22+C(Ω23+ω^(C(Ω22)2),0),0),0)
  • {1` `,,2{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}2`,,} has recursion level C(C(Ω22+C(Ω23+C(C(Ω2,C(Ω22)),C(Ω22)),0),0),0)
  • {1{1,,`1`,,2,,}2{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}2`,,} has recursion level C(C(Ω22+C(Ω23+C(C(Ω222),C(Ω22)),0),0),0)
  • {1{1,,`1{1,,`1,,`2,,}2,,}2{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}2`,,} has recursion level C(C(Ω22+C(Ω23+C(σ3,C(Ω22)),0),0),0)
  • {1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}3`,,} has recursion level C(C(Ω22+C(Ω23+C(Ω2,C(Ω22)),0),0),0)
  • {1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}1,2`,,} has recursion level C(C(Ω22+C(Ω23+C(Ω2+1,σ2),0),0),0)
  • {1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}1`,,2`,,} has recursion level C(C(Ω22+C(Ω23+C(Ω222),0),0),0)
  • {1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}1 {1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}2`,,}2`,,} has recursion level C(C(Ω22+C(Ω23+C(Ω2+C(Ω22),σ2),0),0),0)
  • {1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}1 {1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}1`,,2`,,}2`,,} has recursion level C(C(Ω22+C(Ω23+C(Ω2+C(Ω222),σ2),0),0),0)
  • {1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}1` `,,2`,,} has recursion level C(C(Ω22+C(Ω23+C(Ω232),0),0),0)
  • {1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}1` `,,1 {1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}1` `,,2`,,}2`,,} has recursion level C(C(Ω22+C(Ω23+C(Ω23+C(Ω232),σ2),0),0),0)
  • {1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}1` `,,1` `,,2`,,} has recursion level C(C(Ω22+C(Ω232,0),0),0)
  • {1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}1{1,,`1,2,,}2`,,} has recursion level C(C(Ω22+C(Ω2+C(C(Ω2+1,σ3),σ3),0),0),0)
  • {1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}1{1,,`1`,,2,,}2`,,} has recursion level C(C(Ω22+C(Ω2+C(C(Ω223),σ3),0),0),0)
  • {1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}1{1,,`1` `,,2,,}2`,,} has recursion level C(C(Ω22+C(Ω2+C(C(Ω233),σ3),0),0),0)
  • {1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}1{1,,`1{1{1,,`1` `,,2,,}2` `,,}2,,}2`,,} has recursion level C(C(Ω22+C(Ω2+C(C(Ω2+C(C(Ω233),σ3),σ3),σ3),0),0),0)
  • {1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}1{1,,`1` ` `,,2,,}2`,,} has recursion level C(C(Ω22+C(Ω2+C(σ43),0),0),0)
  • {1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}1{1,,`1` ` `,,1,2,,}2`,,} has recursion level C(C(Ω22+C(Ω2+C(C(Ω22+1,0),σ3),0),0),0)
  • {1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}1{1,,`1` ` `,,1,,2,,}2`,,} has recursion level C(C(Ω22+C(Ω2+C(Ω23),0),0),0)
  • {1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}1{1,,`1{1,,`1,2,,}2,,}2`,,} has recursion level C(C(Ω22+C(Ω2+C(Ω2+1,σ3),0),0),0)
  • {1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}1{1,,`1{1,,`1` `,,2,,}2,,}2`,,} has recursion level C(C(Ω22+C(Ω2+C(Ω233),0),0),0)
  • {1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}1{1,,`1{1,,`1{1,,`1` `,,2,,}2,,}2,,}2`,,} has recursion level C(C(Ω22+C(Ω2+C(Ω2+C(Ω233),σ3),0),0),0)
  • {1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}1{1,,`1{1,,`1,,`2,,}2,,}2`,,} has recursion level C(C(Ω22+C(Ω24,0),0),0)
  • {1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}2`,,} has recursion level C(C(Ω22+C(Ω24+C(Ω23),0),0),0)
  • {1{2,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}2`,,} has recursion level C(C(Ω22+C(Ω24+ω^(C(Ω23)+1),0),0),0)
  • {1` `,,2,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,} has recursion level C(C(Ω22+C(Ω24+ω^(C(Ω23)+σ3),0),0),0)

Generally, the recursion level of {1{1,,`k,,}2,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,} is between C(C(Ω22+C(Ω2k+1,0),0),0) and C(C(Ω22+C(Ω2k+2,0),0),0) (where σ1 = C(Ω22,0), and σn+1 = C(Ω22,σn)), so {1{1,,`1,2,,}2,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(Ω22+1,0),0),0),0).

  • {1{1{1,,`1,2,,}2,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}1,2} has recursion level C(C(Ω22+C(Ω2+C(Ω22+1,0)+1,0),0),0)
  • {1{1{1,,`1,2,,}2,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}1,,1,2} has recursion level C(C(Ω22+C(Ω2+C(Ω22+1,0)+C(Ω2,C(Ω22,0))+1,0),0),0)
  • {1{1{1,,`1,2,,}2,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}1 {1{1,,`1,2,,}2,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}2} has recursion level C(C(Ω22+C(Ω2+C(Ω22+1,0)+C(Ω2+C(Ω22+1,0),C(Ω22,0)),0),0),0)
  • {1,,2{1,,`1,2,,}2,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(Ω22+1,0)+ω^(C(Ω2+C(Ω22+1,0),C(Ω22,0))+C(Ω2,C(Ω22,0))),0),0),0)
  • {1`,,2{1,,`1,2,,}2,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(Ω22+1,0)+ω^(C(Ω2+C(Ω22+1,0),C(Ω22,C(Ω22,0)))+C(Ω22,C(Ω22,0))),0),0),0)
  • {1` `,,2{1,,`1,2,,}2,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(Ω22+1,0)+ω^(C(Ω2+C(Ω22+1,0),C(Ω22,C(Ω22,C(Ω22,0))))+C(Ω22,C(Ω22,C(Ω22,0)))),0),0),0)
  • {1{1,,`1,2,,}3,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(Ω22+1,0)2,0),0),0)
  • {1{1{1,,`2,2,,}2,,`1,2,,}2,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(C(Ω2,C(Ω22+1,0)),C(Ω22+1,0)),0),0),0)
  • {1{1{1,,`1{1,,`1,2,,}2,,}2,,`1,2,,}2,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(C(Ω2+C(Ω22+1,0),C(Ω22+1,0)),C(Ω22+1,0)),0),0),0)
  • {1{1{1,,`1{1{1,,`1{1,,`1,2,,}2,,}2,,`1,2,,}2,,}2,,`1,2,,}2,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(C(Ω2+C(C(Ω2+C(Ω22+1,0),C(Ω22+1,0)),C(Ω22+1,0)),C(Ω22+1,0)),C(Ω22+1,0)),0),0),0)
  • {1{1{1,,`1{1,,`2,2,,}2,,}2,,`1,2,,}2,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(C(Ω22,C(Ω22+1,0)),C(Ω22+1,0)),0),0),0)
  • {1{1{1,,`1{1,,`2,2,,}1,2,,}2,,`1,2,,}2,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(C(Ω22+1,C(Ω22+1,0)),C(Ω22+1,0)),0),0),0)
  • {1{1{1,,`1{1,,`2,2,,}1,1,2,,}2,,`1,2,,}2,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(C(Ω22+2,0),C(Ω22+1,0)),0),0),0)
  • {1{1{1,,`1{1,,`2,2,,}1,,2,,}2,,`1,2,,}2,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(Ω2,C(Ω22+1,0)),0),0),0)
  • {1{1{1,,`1{1,,`3,2,,}2,,}2,,`1,2,,}2,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(C(Ω22,C(Ω22+1,0)),C(Ω2,C(Ω22+1,0))),0),0),0)
  • {1{1{1,,`1{1,,`1,3,,}2,,}2,,`1,2,,}2,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(Ω2+1,C(Ω22+1,0)),0),0),0)
  • {1{1{1,,`1{1,,`1{1,,`1,2,,}2,,}2,,}2,,`1,2,,}2,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(Ω2+C(Ω22+1,0),C(Ω22+1,0)),0),0),0)
  • {1{1{1,,`1{1,,`1{1,,`1{1,,`1,2,,}2,,}2,,}2,,}2,,`1,2,,}2,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(Ω2+C(Ω2+C(Ω22+1,0),C(Ω22+1,0)),C(Ω22+1,0)),0),0),0)
  • {1{1{1,,`1{1,,`1,,`2,,}2,,}2,,`1,2,,}2,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(Ω22,C(Ω22+1,0)),0),0),0)
  • {1{1,,`1,2,,}2{1{1,,`1{1,,`1,,`2,,}2,,}2,,`1,2,,}2,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(Ω22,C(Ω22+1,0))+C(Ω22+1,0),0),0),0)
  • {1{1{1,,`1{1,,`1,,`2,,}2,,}2,,`1,2,,}3,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(Ω22,C(Ω22+1,0))+C(C(Ω22,C(Ω22+1,0)),C(Ω22+1,0)),0),0),0)
  • {1{1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}2{1,,`1,,`2,,}2,,}2,,`1,2,,} 2,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(Ω22,C(Ω22+1,0))+C(C(Ω22,C(Ω22,C(Ω22+1,0))),C(Ω22+1,0)),0),0),0)
  • {1{1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}2,,`1,2,,} 2,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(Ω22,C(Ω22+1,0))+C(Ω2,C(Ω22+1,0)),0),0),0)
  • {1{1,,`1,2,,}2{1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}2,,`1,2,,} 2,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(Ω22,C(Ω22+1,0))+C(Ω2,C(Ω22+1,0))+C(Ω22+1,0),0),0),0)
  • {1{1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}2,,`1,2,,} 3,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(Ω22,C(Ω22+1,0))+C(Ω2,C(Ω22+1,0))2,0),0),0)
  • {1{2{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}2,,`1,2,,} 2,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(Ω22,C(Ω22+1,0))+ω^ω^(C(Ω2,C(Ω22+1,0))+1),0),0),0)
  • {1{1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}3,,`1,2,,} 2,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(Ω22,C(Ω22+1,0))+C(Ω2,C(Ω2,C(Ω22+1,0))),0),0),0)
  • {1{1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}1{1,,`1,2,,}2,,`1,2,,} 2,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(Ω22,C(Ω22+1,0))+C(Ω2+C(Ω22+1,0),C(Ω22+1,0)),0),0),0)
  • {1{1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}1 {1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}2,,`1,2,,}2,,`1,2,,} 2,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(Ω22,C(Ω22+1,0))+C(Ω2+C(Ω2,C(Ω22+1,0)),C(Ω22+1,0)),0),0),0)
  • {1{1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}1{1,,`2,2,,}2,,`1,2,,} 2,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(Ω22,C(Ω22+1,0))2,0),0),0)
  • {1{1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}1{1,,`1{1,,`3,2,,}2,,}2,,`1,2,,} 2,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(C(Ω22,C(Ω22,C(Ω22+1,0))),C(Ω22,C(Ω22+1,0))),0),0),0)
  • {1{1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}1{1,,`1{1,,`4,2,,}2,,}2,,`1,2,,} 2,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(C(Ω22,C(Ω22,C(Ω22+1,0))),C(Ω2,C(Ω22,C(Ω22+1,0)))),0),0),0)
  • {1{1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}1{1,,`1{1,,`1,3,,}2,,}2,,`1,2,,} 2,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(Ω2+1,C(Ω22,C(Ω22+1,0))),0),0),0)
  • {1{1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}1{1,,`1{1,,`1,,`2,,}2,,}2,,`1,2,,} 2,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(Ω22,C(Ω22,C(Ω22+1,0))),0),0),0)
  • {1{1{2,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}2,,`1,2,,} 2,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,} has recursion level C(C(Ω22+C(Ω2+ω^(C(Ω22,C(Ω22,C(Ω22+1,0)))+1),0),0),0)
  • {1{1,,`2,2,,}2,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,} has recursion level C(C(Ω22+C(Ω2+ω^(C(Ω22,C(Ω22,C(Ω22+1,0)))+C(Ω22,C(Ω22+1,0))),0),0),0)
  • {1{1,,`3,2,,}2,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,} has recursion level C(C(Ω22+C(Ω2+ω^(C(Ω22,C(Ω22,C(Ω22,C(Ω22+1,0))))+C(Ω22,C(Ω22,C(Ω22+1,0)))),0),0),0)
  • {1{1,,`1,3,,}2,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(Ω22+1,C(Ω22+1,0)),0),0),0)
  • {1{1,,`1,1,2,,}2,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(Ω22+2,0),0),0),0)
  • {1{1,,`1{1{1{1,,`1,2,,}2,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}2}2,,}2,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(Ω22+C(Ω2+C(Ω22+1,0),0),0),0),0),0)
  • {1{1,,`1,,2,,}2,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,} has recursion level C(C(Ω22+C(Ω22,0),0),0)

Up to here, some approximations make the comparisons above more simple:

  • The double comma approximately corresponds to C(Ω2,C(Ω22,0))
  • `,, approximately corresponds to C(Ω22,C(Ω22,0))
  • {1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}2`,,} approximately corresponds to C(Ω2,C(Ω22,C(Ω22,0)))
  • ` `,, approximately corresponds to C(Ω22,C(Ω22,C(Ω22,0)))
  • {1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}2` `,,} approximately corresponds to C(Ω2,C(Ω22,C(Ω22,C(Ω22,0))))
  • {1,,`1,2,,} approximately corresponds to C(Ω22+1,0)
  • {1,,`2,2,,} approximately corresponds to C(Ω22,C(Ω22+1,0))
  • {1,,`3,2,,} approximately corresponds to C(Ω22,C(Ω22,C(Ω22+1,0)))

Beyond C(Ω22+C(Ω22,0),0)

Now let separator ♦ = {1{1{1,,`1,,2,,}2,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}2} and ♥ = {1{1,,`1♦2,,}2,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,} and let ordinal σ = C(Ω22,0).

  • {1,,1{1{1♥2}2,,1{1,,1,,2}2}2{1,,1,,2}2} has recursion level C(C(Ω22+σ,0),0)
  • {1,,1{1{1♥2}2,,1{1,,1,,2}2}1,2{1,,1,,2}2} has recursion level C(C(Ω22+σ+1,0),0)
  • {1,,1{2{1,,1{1♥2}2}2,,1{1,,1,,2}2}2{1,,1,,2}2} has recursion level C(C(Ω22+σ+ω^(C(C(Ω22+σ,0),C(Ω2+σ,0))+1),0),0)
  • {1,,2{1♥2}2} has recursion level C(C(Ω22+σ+C(C(Ω2,C(Ω22+σ,0)),C(Ω2+σ,0)),0),0)
  • {1,,1{1,,1{1♥2}2}2{1♥2}2} has recursion level C(C(Ω22+σ+C(C(Ω22+σ,C(Ω22+σ,0)),C(Ω2+σ,0)),0),0)
  • {1,,1{1,,1{1♥2}2}1,2{1♥2}2} has recursion level C(C(Ω22+σ+C(C(Ω22+σ+1,0),C(Ω2+σ,0)),0),0)
  • {1,,1{1,,1{1♥2}2}1{1,,2{1,,1,,2}2}2{1♥2}2} has recursion level C(C(Ω22+σ+C(Ω2,C(Ω2+σ,0)),0),0)
  • {1,,1{1,,1{1♥2}2}1{1,,1{1,,1,,2}1,2}2{1♥2}2} has recursion level C(C(Ω22+σ+C(Ω2+σ+1,0),0),0)
  • {1,,1{1,,1{1♥2}2}1{1,,1{1,,1{1,,2,,2}1,2,,2}2}2{1♥2}2} has recursion level C(C(Ω22+σ+C(Ω2+C(C(Ω22+1,0),σ),0),0),0)
  • {1,,1{1,,1{1♥2}2}1{1,,1{1♥2}2}2{1♥2}2} has recursion level C(C(Ω22+σ+C(Ω2+C(C(Ω22+σ,0),σ),0),0),0)
  • {1,,1{1,,1,,2}2{1♥2}2} has recursion level C(C(Ω22+σ+C(σ,C(Ω2+C(C(Ω22+σ,0),σ),0)),0),0)
  • {1,,1{1♥2}3} has recursion level C(C(Ω22+σ+C(C(Ω22+σ,0),C(Ω2+C(C(Ω22+σ,0),σ),0)),0),0)
  • {1,,1{1♥2}1,2} has recursion level C(C(Ω22+σ+C(Ω2+C(C(Ω22+σ,0),σ)+1,0),0),0)
  • {1{1,,1,,2}2♥2} has recursion level C(C(Ω22+σ+C(Ω2+ω^(C(C(Ω22+σ,0),σ)+σ),0),0),0)
  • {1{1,,1{1,,2,,2}1{1,,1,,2}2,,2}2,,1{1,,2,,2}1{1{1♥2}2,,1,,2}2,,2} has recursion level C(C(Ω22+σ+C(Ω2+C(C(Ω22+σ,0)+C(Ω22+C(Ω2+C(Ω2,σ),0),0),σ),0),0),0)
  • {1{1,,1{1,,2,,2}1{1{1,,1{1,,1,,3}2,,2}2,,1,,2}2,,2}2,,1{1,,2,,2}1{1{1♥2}2,,1,,2}2,,2} has recursion level C(C(Ω22+σ+C(Ω2+C(C(Ω22+σ,0)+C(Ω22+C(σ,C(Ω2+C(Ω2,σ),0)),0),σ),0),0),0)
  • {1{1,,1{1,,2,,2}1{1{1,,1,,1,2}2,,1,,2}2,,2}2,,1{1,,2,,2}1{1{1♥2}2,,1,,2}2,,2} has recursion level C(C(Ω22+σ+C(Ω2+C(C(Ω22+σ,0)+C(Ω22+C(Ω2+C(Ω2,σ)+1,0),0),σ),0),0),0)
  • {1{1,,1{1,,2,,2}1{1{1{1,,`1{1,,`1,,`2,,}2,,}2}2,,1,,2}2,,2}2,,1{1,,2,,2}1{1{1♥2}2,,1,,2}2,,2} has recursion level C(C(Ω22+σ+C(Ω2+C(C(Ω22+σ,0)+C(Ω22+C(Ω2+C(Ω22,σ),0),0),σ),0),0),0)
  • {1{1,,1{1,,2,,2}1{1{1♥2}2,,1,,2}2,,2}2,,1{1,,2,,2}1{1{1♥2}2,,1,,2}2,,2} has recursion level C(C(Ω22+σ+C(Ω2+C(C(Ω22+σ,0)2,σ),0),0),0)
  • {1,,2{1,,2,,2}1{1{1♥2}2,,1,,2}2,,2} has recursion level C(C(Ω22+σ+C(Ω2+C(C(Ω2,C(Ω22+σ,0)),σ),0),0),0)
  • {1,,1{1,,2,,2}1{1{1♥2}2,,1,,2}1,2,,2} has recursion level C(C(Ω22+σ+C(Ω2+C(C(Ω22+σ+1,0),σ),0),0),0)
  • {1,,1{1,,2,,2}1{1{1♥2}2,,1,,2}1{1,,1,,2}2,,2} has recursion level C(C(Ω22+σ+C(Ω2+C(Ω2,σ),0),0),0)
  • {1{1,,1,,1,2}2♥2} has recursion level C(C(Ω22+σ+C(Ω2+C(Ω2,σ)+1,0),0),0)
  • {1{1{1,,`1`,,2,,}2}2♥2} has recursion level C(C(Ω22+σ+C(Ω2+C(C(Ω22,σ),C(Ω2,σ)),0),0),0)
  • {1{1{1,,`1`,,1{1♥2}2,,}2}2♥2} has recursion level C(C(Ω22+σ+C(Ω2+C(C(Ω22+σ,0),C(Ω2,σ)),0),0),0)
  • {1{1{1,,`1`,,1,,2,,}2}2♥2} has recursion level C(C(Ω22+σ+C(Ω2+C(Ω2,C(Ω2,σ)),0),0),0)
  • {1{1{1,,`1` `,,2,,}2}2♥2} has recursion level C(C(Ω22+σ+C(Ω2+C(C(Ω22,σ),C(Ω2,C(Ω2,σ))),0),0),0)
  • {1{1{1,,`1{1,,`1,2,,}2,,}2}2♥2} has recursion level C(C(Ω22+σ+C(Ω2+C(Ω2+1,σ),0),0),0)
  • {1{1{1,,`1{1,,`1,,`2,,}2,,}2}2♥2} has recursion level C(C(Ω22+σ+C(Ω2+C(Ω22,σ),0),0),0)
  • {1{1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}2}2♥2} has recursion level C(C(Ω22+σ+C(Ω2+C(Ω22,σ)+C(Ω2,C(Ω2,σ)),0),0),0)
  • {1{1{1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}1`,,1`,,2,,}2}2♥2} has recursion level C(C(Ω22+σ+C(Ω2+C(Ω22,σ)2,0),0),0)
  • {1{1{1{1,,`1,2,,}2,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}2}2♥2} has recursion level C(C(Ω22+σ+C(Ω2+C(Ω22+1,0),0),0),0)
  • {1{1♥2}2♥2} has recursion level C(C(Ω22+σ+C(Ω2+C(Ω22+σ,0),0),0),0)
  • {1,,1,,2♥2} has recursion level C(C(Ω22+σ+C(σ,C(Ω2+C(Ω22+σ,0),0)),0),0)
  • {1,,1{1,,2,,2♥2}1,2,,2♥2} has recursion level C(C(Ω22+σ+C(Ω2+C(C(Ω22+1,0),σ),C(Ω2+C(Ω22+σ,0),0)),0),0)
  • {1,,1{1,,2,,2♥2}1{1,,1,,2♥2}2,,2♥2} has recursion level C(C(Ω22+σ+C(Ω2+C(C(Ω22+C(Ω2+C(Ω2,σ),C(Ω2+C(Ω22+σ,0),0)),0),σ),C(Ω2+C(Ω22+σ,0),0)),0),0)
  • {1,,1,,1,2♥2} has recursion level C(C(Ω22+σ+C(Ω2+C(C(Ω22+C(Ω2+C(Ω2,σ)+1,C(Ω2+C(Ω22+σ,0),0)),0),σ),C(Ω2+C(Ω22+σ,0),0)),0),0)
  • {1{1{1,,`1,2,,}2,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}2♥2} has recursion level C(C(Ω22+σ+C(Ω2+C(C(Ω22+C(Ω2+C(Ω22+1,0),C(Ω2+C(Ω22+σ,0),0)),0),σ),C(Ω2+C(Ω22+σ,0),0)),0),0)
  • {1♥3} has recursion level C(C(Ω22+σ+C(Ω2+C(C(Ω22+σ,0),σ),C(Ω2+C(Ω22+σ,0),0)),0),0)
  • {1{1♥3}2♥3} has recursion level C(C(Ω22+σ+C(Ω2+C(Ω22+σ,0),C(Ω2+C(Ω22+σ,0),0)),0),0)
  • {1♥1,2} has recursion level C(C(Ω22+σ+C(Ω2+C(Ω22+σ,0)+1,0),0),0)
  • {1♥1,,1,2} has recursion level C(C(Ω22+σ+C(Ω2+C(Ω22+σ,0)+C(Ω2,σ)+1,0),0),0)
  • {1♥1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,2{1,,`1,,`2,,}2,,}2} has recursion level C(C(Ω22+σ+C(Ω2+C(Ω22+σ,0)+C(C(Ω22+1,0),C(Ω2,σ)),0),0),0)
  • {1♥1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1{1♥2}2{1,,`1,,`2,,}2,,}2} has recursion level C(C(Ω22+σ+C(Ω2+C(Ω22+σ,0)+C(C(Ω22+C(Ω2+C(Ω22+σ,0),0),0),C(Ω2,σ)),0),0),0)
  • {1♥1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}2} has recursion level C(C(Ω22+σ+C(Ω2+C(Ω22+σ,0)+C(C(Ω22+σ,0),C(Ω2,σ)),0),0),0) (this time, the σ in C(Ω2+C(Ω22+σ,0)+C(C(Ω22+σ,0),C(Ω2,σ)),0) works as a 1-separator)
  • {1♥1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1 {1♥1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}2} 2{1,,`1,,`2,,}2,,}1,2} has recursion level C(C(Ω22+σ+C(Ω2+C(Ω22+σ,0)+C(C(Ω22+σ,0),C(Ω2,σ))+1,0),0),0)
  • {1♥1{2{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1 {1♥1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}2} 2{1,,`1,,`2,,}2,,}2,,}2} has recursion level C(C(Ω22+σ+C(Ω2+C(Ω22+σ,0)+ω^(C(C(Ω22+σ,0),C(Ω2,σ))+1),0),0),0)
  • {1♥1{1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1 {1♥1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}2} 2{1,,`1,,`2,,}2,,}1,2,,}2} has recursion level C(C(Ω22+σ+C(Ω2+C(Ω22+σ,0)+C(C(Ω22+σ,0)+1,C(Ω2,σ)),0),0),0)
  • {1♥1{1,,`2{1,,`1{1,,`1,,`2,,}2,,}1 {1♥1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}2} 2{1,,`1,,`2,,}2,,}2} has recursion level C(C(Ω22+σ+C(Ω2+C(Ω22+σ,0)+C(C(Ω2,C(Ω22+σ,0)),C(Ω2,σ)),0),0),0)
  • {1♥1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}2 {1♥1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}2} 2{1,,`1,,`2,,}2,,}2} has recursion level C(C(Ω22+σ+C(Ω2+C(Ω22+σ,0)+C(C(Ω22,C(Ω22+σ,0)),C(Ω2,σ)),0),0),0)
  • {1♥1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1 {1♥1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}2} 1,2{1,,`1,,`2,,}2,,}2} has recursion level C(C(Ω22+σ+C(Ω2+C(Ω22+σ,0)+C(C(Ω22+σ+1,0),C(Ω2,σ)),0),0),0)
  • {1♥1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1 {1♥1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}2}1 {1♥1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}2} 2{1,,`1,,`2,,}2,,}2} has recursion level C(C(Ω22+σ+C(Ω2+C(Ω22+σ,0)+C(Ω2,C(Ω2,σ)),0),0),0)
  • {1♥1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}1,2} has recursion level C(C(Ω22+σ+C(Ω2+C(Ω22+σ,0)+C(Ω2,C(Ω2,σ))+1,0),0),0) (however, the extra 1-separator doesn’t affect the strength much)
  • {1♥1{1{1,,`1,2,,}2,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}2} has recursion level C(C(Ω22+σ+C(Ω2+C(Ω22+σ,0)+C(Ω2+C(Ω22+1,0),σ),0),0),0)
  • {1♥1♥2} has recursion level C(C(Ω22+σ+C(Ω2+C(Ω22+σ,0)+C(Ω2+C(Ω22+σ,0),σ),0),0),0)
  • {1{2♥2,,}2} has recursion level C(C(Ω22+σ+C(Ω2+C(Ω22+σ,0)+ω^(C(Ω2+C(Ω22+σ,0),σ)+1),0),0),0)
  • {1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,2{1,,`1,,`2,,}2,,}2♥2,,} has recursion level C(C(Ω22+σ+C(Ω2+C(Ω22+σ,0)+C(C(Ω22+1,0),C(Ω2+C(Ω22+σ,0),σ)),0),0),0)
  • {1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}2♥2,,} has recursion level C(C(Ω22+σ+C(Ω2+C(Ω22+σ,0)+C(C(Ω22+σ,0),C(Ω2+C(Ω22+σ,0),σ)),0),0),0)
  • {1{1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}2♥2,,}1,2} has recursion level C(C(Ω22+σ+C(Ω2+C(Ω22+σ,0)+C(Ω2,C(Ω2+C(Ω22+σ,0),σ))+1,0),0),0)
  • {1♥1,2,,} has recursion level C(C(Ω22+σ+C(Ω2+C(Ω22+σ,0)+C(Ω2+C(Ω22+σ,0)+1,σ),0),0),0)
  • {1♥1`,,2,,} has recursion level C(C(Ω22+σ+C(Ω2+C(Ω22+σ,0)+C(Ω22,σ),0),0),0)
  • {1{1,,`1,2,,}2{1,,`1♦2,,}2,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,} has recursion level C(C(Ω22+σ+C(Ω2+C(Ω22+σ,0)+C(Ω22+1,0),0),0),0)
  • {1{1,,`1♦2,,}3,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,} has recursion level C(C(Ω22+σ+C(Ω2+C(Ω22+σ,0)2,0),0),0)
  • {1{1,,`1,2♦2,,}2,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,} has recursion level C(C(Ω22+σ+C(Ω2+C(Ω22+1,C(Ω22+σ,0)),0),0),0)
  • {1{1,,`1♦3,,}2,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,} has recursion level C(C(Ω22+σ+C(Ω2+C(Ω22+σ,C(Ω22+σ,0)),0),0),0)
  • {1{1,,`1♦1,2,,}2,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,} has recursion level C(C(Ω22+σ+C(Ω2+C(Ω22+σ+1,0),0),0),0)
  • {1{1,,`1♦1♦2,,}2,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,} has recursion level C(C(Ω22+σ2,0),0)
  • {1,,2{1{1,,`1,,2,,}2,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}2} has recursion level C(C(Ω22+C(C(Ω2,σ),σ),0),0)
  • {1,,1,2{1{1,,`1,,2,,}2,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}2} has recursion level C(C(Ω22+C(C(Ω2+1,σ),σ),0),0)
  • {1,,1,,2{1{1,,`1,,2,,}2,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}2} has recursion level C(C(Ω22+C(C(Ω22,σ),σ),0),0)

At this stage, C(Ω2+____,0) ordinals correspond to {1{1{1,,`1 A 2,,}2,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}2} with lv(A) < lv(,,); and C(Ω22,0) is the supremum of them, so it corresponds to {1{1{1,,`1,,2,,}2,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}2}. When it reduces, the double comma (red) is the 2-separator, {1{1{1,,`1,,2,,}2,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}2} is where it drops down to (1-separator).

Now let separator ◊ = {1{1,,`1,,2,,}2,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,} and let ordinal σ = C(Ω22,0).

  • {1{1,,1{1,,1,,2◊2}2◊2}2◊2} has recursion level C(C(Ω22+C(C(Ω22,σ),σ),0),0)
  • {1{1,,2◊2}2{1,,1{1,,1,,2◊2}2◊2}2◊2} has recursion level C(C(Ω22+C(C(Ω2,σ),C(C(Ω22,σ),σ)),0),0)
  • {1{1,,1{1,,1,,2◊2}2◊2}3◊2} has recursion level C(C(Ω22+C(C(Ω22,σ),C(C(Ω22,σ),σ)),0),0)
  • {1{1,,1{1,,1,,2◊2}2◊2}1,2◊2} has recursion level C(C(Ω22+C(C(Ω22,σ)+1,σ),0),0)
  • {1{1,,1{1,,1,,2◊2}2◊2}2,,1{1,,1,,2◊2}2◊2} has recursion level C(C(Ω22+C(C(Ω22,σ)2,σ),0),0)
  • {1,,2{1,,1,,2◊2}2◊2} has recursion level C(C(Ω22+C(C(Ω2,C(Ω22,σ)),σ),0),0)
  • {1,,1{1,,1,2◊2}2{1,,1,,2◊2}2◊2} has recursion level C(C(Ω22+C(C(Ω2+C(Ω2+1,σ),C(Ω22,σ)),σ),0),0)
  • {1,,1{1,,1{1,,1,,2◊2}2◊2}2{1,,1,,2◊2}2◊2} has recursion level C(C(Ω22+C(C(Ω22,C(Ω22,σ)),σ),0),0)
  • {1,,1{1,,1{1,,1,,2◊2}2◊2}1,2{1,,1,,2◊2}2◊2} has recursion level C(C(Ω22+C(C(Ω22+1,0),σ),0),0)
  • {1,,1{1,,1{1,,1,,2◊2}2◊2}1{1{1{1,,`1{1◊2}2,,}2,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}2}2 {1,,1,,2◊2}2◊2} has recursion level C(C(Ω22+C(C(Ω22+C(Ω2+C(Ω22+σ,0),0),0),σ),0),0)
  • {1,,1{1,,1{1,,1,,2◊2}2◊2}1{1◊2}2{1,,1,,2◊2}2◊2} has recursion level C(C(Ω22+C(C(Ω22+σ,0),σ),0),0)
  • {1,,1{1,,1{1,,1,,2◊2}2◊2}1{1{1,,1{1,,1{1,,1,,2◊2}2◊2}1{1◊2}2{1,,1,,2◊2}2◊2}2◊2}2{1,,1,,2◊2}2◊2} has recursion level C(C(Ω22+C(C(Ω22+C(C(Ω22+σ,0),σ),0),σ),0),0)
  • {1,,1{1,,1{1,,1,,2◊2}2◊2}1{1,,2◊2}2{1,,1,,2◊2}2◊2} has recursion level C(C(Ω22+C(Ω2,σ),0),0)

Now let separator ◊ = {1{1,,`1,,2,,}2,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,} and ♦ = {1,,1{1,,1{1,,1,,2◊2}2◊2}1{1,,2◊2}2{1,,1,,2◊2}2◊2} and ♥ = {1{1,,`1♦2,,}2,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,} and let ordinal σ = C(Ω22,0).

  • {1,,1{1{1♥2}2,,1{1,,1,,2}2}1,2{1,,1,,2}2} has recursion level C(C(Ω22+C(Ω2,σ)+1,0),0)
  • {1,,1{1{1♥2}2,,1{1,,1,,2}2}1{1{1♥2}2,,1{1,,1,,2}2}2{1,,1,,2}2} has recursion level C(C(Ω22+C(Ω2,σ)+C(C(Ω22+C(Ω2,σ),0),C(Ω2+σ,0)),0),0)
  • {1,,1{2{1,,1{1♥2}2}2,,1{1,,1,,2}2}2{1,,1,,2}2} has recursion level C(C(Ω22+C(Ω2,σ)+ω^(C(C(Ω22+C(Ω2,σ),0),C(Ω2+σ,0))+1),0),0)
  • {1,,2{1♥2}2} has recursion level C(C(Ω22+C(Ω2,σ)+C(C(Ω2,C(Ω22+C(Ω2,σ),0)),C(Ω2+σ,0)),0),0)
  • {1,,1{1,,1{1♥2}2}1,2{1♥2}2} has recursion level C(C(Ω22+C(Ω2,σ)+C(C(Ω22+C(Ω2,σ)+1,0),C(Ω2+σ,0)),0),0)
  • {1,,1{1,,1{1♥2}2}1{1,,2{1,,1,,2}2}2{1♥2}2} has recursion level C(C(Ω22+C(Ω2,σ)+C(Ω2,C(Ω2+σ,0)),0),0)
  • {1,,1{1,,1{1♥2}2}1{1,,1{1{1{1,,`1{1◊2}2,,}2,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}2}2}2{1♥2}2} has recursion level C(C(Ω22+C(Ω2,σ)+C(Ω2+C(C(Ω22+σ,0),σ),0),0),0)
  • {1,,1{1,,1{1♥2}2}1{1,,1{1♥2}2}2{1♥2}2} has recursion level C(C(Ω22+C(Ω2,σ)+C(Ω2+C(C(Ω22+C(Ω2,σ),0),σ),0),0),0)
  • {1,,1{1♥2}1,2} has recursion level C(C(Ω22+C(Ω2,σ)+C(Ω2+C(C(Ω22+C(Ω2,σ),0),σ)+1,0),0),0)
  • {1{1,,1,,2}2♥2} has recursion level C(C(Ω22+C(Ω2,σ)+C(Ω2+ω^(C(C(Ω22+C(Ω2,σ),0),σ)+1),0),0),0)
  • {1{1,,1,,1,2}2♥2} has recursion level C(C(Ω22+C(Ω2,σ)+C(Ω2+C(Ω2,σ)+1,0),0),0)
  • {1{1{1,,`1`,,1{1♥2}2,,}2}2♥2} has recursion level C(C(Ω22+C(Ω2,σ)+C(Ω2+C(C(Ω22+C(Ω2,σ),0),C(Ω2,σ)),0),0),0)
  • {1{1{1,,`1`,,1,,2,,}2}2♥2} has recursion level C(C(Ω22+C(Ω2,σ)+C(Ω2+C(Ω2,C(Ω2,σ)),0),0),0)
  • {1{1{1{1,,`1{1◊2}2,,}2,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}2}2♥2} has recursion level C(C(Ω22+C(Ω2,σ)+C(Ω2+C(Ω22+σ,0),0),0),0)
  • {1{1{1{1,,`1{1◊2}2,,}2,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}1,2}2♥2} has recursion level C(C(Ω22+C(Ω2,σ)+C(Ω2+C(Ω22+σ,0)+1,0),0),0)
  • {1 {1{1{1,,`1{1◊2}2,,}2,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}1 {1,,`1{1,,`1{1,,`1,,`2,,}2,,}1 {1{1{1,,`1{1◊2}2,,}2,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}1 {1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}2} 2{1,,`1,,`2,,}2,,}2} 2♥2} has recursion level C(C(Ω22+C(Ω2,σ)+C(Ω2+C(Ω22+σ,0)+C(C(Ω22+σ,0),C(Ω2,σ)),0),0),0)
  • {1 {1{1{1,,`1{1◊2}2,,}2,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}1 {1,,`1{1,,`1{1,,`1,,`2,,}2,,}1 {1{1{1,,`1{1◊2}2,,}2,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}1 {1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}2} 1,2{1,,`1,,`2,,}2,,}2} 2♥2} has recursion level C(C(Ω22+C(Ω2,σ)+C(Ω2+C(Ω22+σ,0)+C(C(Ω22+σ+1,0),C(Ω2,σ)),0),0),0)
  • {1 {1{1{1,,`1{1◊2}2,,}2,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}1 {1,,`1{1,,`1{1,,`1,,`2,,}2,,}1 {2{1{1,,`1{1◊2}2,,}2,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}1 {1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}2} 2{1,,`1,,`2,,}2,,}2} 2♥2} has recursion level C(C(Ω22+C(Ω2,σ)+C(Ω2+C(Ω22+σ,0)+C(C(Ω22+ω^(σ+1),0),C(Ω2,σ)),0),0),0)
  • {1 {1{1{1,,`1{1◊2}2,,}2,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}2}2 {1{1,,`1{1,,1{1,,1{1,,1,,2◊2}2◊2}1{1,,2◊2}2{1,,1,,2◊2}2◊2}2,,} 2,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}2} has recursion level C(C(Ω22+C(Ω2,σ)+C(Ω2+C(Ω22+σ,0),0),0),0)
  • {1 {1{1{1,,`1{1,,1{1,,1{1,,1,,2◊2}2◊2}1{1,,2◊2}2{1,,1,,2◊2}2◊2} 2,,}2,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}2}2 {1{1,,`1{1,,1{1,,1{1,,1,,2◊2}2◊2}1{1,,2◊2}2{1,,1,,2◊2}2◊2}2,,} 2,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}2} has recursion level C(C(Ω22+C(Ω2,σ)+C(Ω2+C(Ω22+C(Ω2,σ),0),0),0),0)
  • {1,,1,,2{1{1,,`1{1,,1{1,,1{1,,1,,2◊2}2◊2}1{1,,2◊2}2{1,,1,,2◊2}2◊2}2,,} 2,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}2} has recursion level C(C(Ω22+C(Ω2,σ)+C(σ,C(Ω2+C(Ω22+C(Ω2,σ),0),0)),0),0)
  • {1{1{1,,`1{1,,1{1,,1{1,,1,,2◊2}2◊2}1{1,,2◊2}2{1,,1,,2◊2}2◊2}2,,} 2,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}1,2} has recursion level C(C(Ω22+C(Ω2,σ)+C(Ω2+C(Ω22+C(Ω2,σ),0)+1,0),0),0)
  • {1{1,,`1{1,,1{1,,1{1,,1,,2◊2}2◊2}1{1,,2◊2}2{1,,1,,2◊2}2◊2}1,2,,} 2,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,} has recursion level C(C(Ω22+C(Ω2,σ)+C(Ω2+C(Ω22+C(Ω2,σ)+1,0),0),0),0)
  • {1{1,,`1{1,,1{1,,1{1,,1,,2◊2}2◊2}1{1,,2◊2}2{1,,1,,2◊2}2◊2}1{1◊2}2,,} 2,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,} has recursion level C(C(Ω22+C(Ω2,σ)+σ,0),0)
  • {1{1,,`1 {1,,1{1,,1{1,,1,,2◊2}2◊2}1{1,,2◊2}2{1,,1,,2◊2}2◊2}1 {1,,1{1,,1{1,,1,,2◊2}2◊2}1,2{1,,1,,2◊2}2◊2} 2,,}2,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,} has recursion level C(C(Ω22+C(Ω2,σ)+C(C(Ω22+1,0),σ),0),0)
  • {1{1,,`1 {1,,1{1,,1{1,,1,,2◊2}2◊2}1{1,,2◊2}2{1,,1,,2◊2}2◊2}1 {1,,1{1,,1{1,,1,,2◊2}2◊2}1{1◊2}2{1,,1,,2◊2}2◊2} 2,,}2,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,} has recursion level C(C(Ω22+C(Ω2,σ)+C(C(Ω22+σ,0),σ),0),0)
  • {1{1,,`1 {1,,1{1,,1{1,,1,,2◊2}2◊2}1{1,,2◊2}2{1,,1,,2◊2}2◊2}1 {1,,1{1,,1{1,,1,,2◊2}2◊2}1{1,,2◊2}2{1,,1,,2◊2}2◊2} 2,,}2,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,} has recursion level C(C(Ω22+C(Ω2,σ)+C(C(Ω22+C(Ω2,σ),0),σ),0),0)
  • {1{1,,2◊2}2{1,,1{1,,1{1,,1,,2◊2}2◊2}1{1,,2◊2}2{1,,1,,2◊2}2◊2}2◊2} has recursion level C(C(Ω22+C(Ω2,σ)+C(C(Ω2,σ),C(C(Ω22+C(Ω2,σ),0),σ)),0),0)
  • {1{1,,1{1,,1{1,,1,,2◊2}2◊2}1,2{1,,1,,2◊2}2◊2}2 {1,,1{1,,1{1,,1,,2◊2}2◊2}1{1,,2◊2}2{1,,1,,2◊2}2◊2}2◊2} has recursion level C(C(Ω22+C(Ω2,σ)+C(C(Ω22+1,0),C(C(Ω22+C(Ω2,σ),0),σ)),0),0)
  • {1{1,,1{1,,1{1,,1,,2◊2}2◊2}1{1◊2}2{1,,1,,2◊2}2◊2}2 {1,,1{1,,1{1,,1,,2◊2}2◊2}1{1,,2◊2}2{1,,1,,2◊2}2◊2}2◊2} has recursion level C(C(Ω22+C(Ω2,σ)+C(C(Ω22+σ,0),C(C(Ω22+C(Ω2,σ),0),σ)),0),0)
  • {1{1,,1{1,,1{1,,1,,2◊2}2◊2}1{1,,2◊2}2{1,,1,,2◊2}2◊2}3◊2} has recursion level C(C(Ω22+C(Ω2,σ)+C(C(Ω22+C(Ω2,σ),0),C(C(Ω22+C(Ω2,σ),0),σ)),0),0)
  • {1{1,,1{1,,1{1,,1,,2◊2}2◊2}1{1,,2◊2}2{1,,1,,2◊2}2◊2}1,2◊2} has recursion level C(C(Ω22+C(Ω2,σ)+C(C(Ω22+C(Ω2,σ),0)+1,σ),0),0)
  • {1,,2{1,,1{1,,1,,2◊2}2◊2}1{1,,2◊2}2{1,,1,,2◊2}2◊2} has recursion level C(C(Ω22+C(Ω2,σ)+C(C(Ω2,C(Ω22+C(Ω2,σ),0)),σ),0),0)
  • {1,,1{1,,1{1,,1,,2◊2}2◊2}2{1,,2◊2}2{1,,1,,2◊2}2◊2} has recursion level C(C(Ω22+C(Ω2,σ)+C(C(Ω22,C(Ω22+C(Ω2,σ),0)),σ),0),0)
  • {1,,1{1,,1{1,,1,,2◊2}2◊2}1{1,,2◊2}1,2{1,,1,,2◊2}2◊2} has recursion level C(C(Ω22+C(Ω2,σ)+C(C(Ω22+C(Ω2,σ)+1,0),σ),0),0)
  • {1,,1{1,,1{1,,1,,2◊2}2◊2}1{1,,2◊2}1{1,,2◊2}2{1,,1,,2◊2}2◊2} has recursion level C(C(Ω22+C(Ω2,σ)2,0),0)
  • {1,,1{1,,1{1,,1,,2◊2}2◊2}1{1{1,,3◊2}2,,2◊2}2{1,,1,,2◊2}2◊2} has recursion level C(C(Ω22+C(C(Ω2,C(Ω2,σ)),C(Ω2,σ)),0),0)
  • {1,,1{1,,1{1,,1,,2◊2}2◊2}1{1,,3◊2}2{1,,1,,2◊2}2◊2} has recursion level C(C(Ω22+C(Ω2,C(Ω2,σ)),0),0)
  • {1,,1{1,,1{1,,1,,2◊2}2◊2}1{1,,1,2◊2}2{1,,1,,2◊2}2◊2} has recursion level C(C(Ω22+C(Ω2+1,σ),0),0)
  • {1,,1{1,,1{1,,1,,2◊2}2◊2}1{1,,1 {1{1{1,,`1{1◊2}2,,}2,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}2} 2◊2}2{1,,1,,2◊2}2◊2} has recursion level C(C(Ω22+C(Ω2+C(Ω2+C(Ω22+σ,0),0),σ),0),0)
  • {1,,1{1,,1{1,,1,,2◊2}2◊2}1{1,,1 {1{1{1,,`1 {1,,1{1,,1{1,,1,,2◊2}2◊2}1{1,,2◊2}2{1,,1,,2◊2}2◊2} 2,,}2,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}2} 2◊2}2{1,,1,,2◊2}2◊2} has recursion level C(C(Ω22+C(Ω2+C(Ω2+C(Ω22+C(Ω2,σ),0),0),σ),0),0)
  • {1,,1{1,,1{1,,1,,2◊2}2◊2}1{1,,1{1◊2}2◊2}2{1,,1,,2◊2}2◊2} has recursion level C(C(Ω22+C(Ω2+σ,σ),0),0)
  • {1,,1{1,,1{1,,1,,2◊2}2◊2}1{1,,1 {1{1,,1{1,,1{1,,1,,2◊2}2◊2}1{1◊2}2{1,,1,,2◊2}2◊2}2◊2} 2◊2}2{1,,1,,2◊2}2◊2} has recursion level C(C(Ω22+C(Ω2+C(C(Ω22+σ,0),σ),σ),0),0)
  • {1,,1{1,,1{1,,1,,2◊2}2◊2}1{1,,1 {1{1,,1{1,,1{1,,1,,2◊2}2◊2}1{1,,2◊2}2{1,,1,,2◊2}2◊2}2◊2} 2◊2}2{1,,1,,2◊2}2◊2} has recursion level C(C(Ω22+C(Ω2+C(C(Ω22+C(Ω2,σ),0),σ),σ),0),0)
  • {1,,1{1,,1{1,,1,,2◊2}2◊2}1{1,,1{1,,2◊2}2◊2}2{1,,1,,2◊2}2◊2} has recursion level C(C(Ω22+C(Ω2+C(Ω2,σ),σ),0),0)
  • {1,,1{1,,1{1,,1,,2◊2}2◊2}1{1,,1{1,,1,2◊2}2◊2}2{1,,1,,2◊2}2◊2} has recursion level C(C(Ω22+C(Ω2+C(Ω2+1,σ),σ),0),0)
  • {1,,1{1,,1{1,,1,,2◊2}2◊2}1{1,,1{1,,1{1,,1,2◊2}2◊2}2◊2}2{1,,1,,2◊2}2◊2} has recursion level C(C(Ω22+C(Ω2+C(Ω2+C(Ω2+1,σ),σ),σ),0),0)
  • {1,,1{1,,1{1,,1,,2◊2}2◊2}1{1,,1{1,,1,,2◊2}2◊2}2{1,,1,,2◊2}2◊2} has recursion level C(C(Ω22+C(Ω2+C(Ω22,σ),σ),0),0)
  • {1,,1{2,,1{1,,1,,2◊2}2◊2}2{1,,1,,2◊2}2◊2} has recursion level C(C(Ω22+ω^(C(Ω2+C(Ω22,σ),σ)+1),0),0)
  • {1,,1{1{1,,2{1,,1,,2◊2}2◊2}2,,1{1,,1,,2◊2}2◊2}2{1,,1,,2◊2}2◊2} has recursion level C(C(Ω22+C(C(Ω2,C(Ω2+C(Ω22,σ),σ)),C(Ω2+C(Ω22,σ),σ)),0),0)
  • {1,,1{1{1,,1{1,,1{1,,1,,2◊2}2◊2}2{1,,1,,2◊2}2◊2}2,,1{1,,1,,2◊2}2◊2}2{1,,1,,2◊2}2◊2} has recursion level C(C(Ω22+C(C(Ω2+C(Ω2+C(Ω22,σ),σ),C(Ω2+C(Ω22,σ),σ)),C(Ω2+C(Ω22,σ),σ)),0),0)
  • {1,,1{1,,2{1,,1,,2◊2}2◊2}2{1,,1,,2◊2}2◊2} has recursion level C(C(Ω22+C(C(Ω2+C(Ω2,C(Ω2+C(Ω22,σ),σ)),C(Ω2+C(Ω22,σ),σ)),C(Ω2+C(Ω22,σ),σ)),0),0)
  • {1,,1{1,,1,,2◊2}3◊2} has recursion level C(C(Ω22+C(C(Ω22,σ),C(Ω2+C(Ω22,σ),σ)),0),0)
  • {1,,1{1,,1{1,,1,,2◊2}3◊2}1{1,,1{1,,1,,2◊2}3◊2}2{1,,1,,2◊2}3◊2} has recursion level C(C(Ω22+C(Ω2+C(Ω22,σ),C(Ω2+C(Ω22,σ),σ)),0),0)
  • {1,,1{1,,1,,2◊2}4◊2} has recursion level C(C(Ω22+C(C(Ω22,σ),C(Ω2+C(Ω22,σ),C(Ω2+C(Ω22,σ),σ))),0),0)
  • {1,,1{1,,1,,2◊2}1,2◊2} has recursion level C(C(Ω22+C(Ω2+C(Ω22,σ)+1,σ),0),0)
  • {1,,1{1,,1,,2◊2}1{1◊2}2◊2} has recursion level C(C(Ω22+C(Ω2+C(Ω22,σ)+σ,σ),0),0)
  • {1,,1{1,,1,,2◊2}1{1,,2◊2}2◊2} has recursion level C(C(Ω22+C(Ω2+C(Ω22,σ)+C(Ω2,σ),σ),0),0)
  • {1,,1{1,,1,,2◊2}1{1,,1{1,,1,,2◊2}2◊2}2◊2} has recursion level C(C(Ω22+C(Ω2+C(Ω22,σ)+C(Ω2+C(Ω22,σ),σ),σ),0),0)
  • {1,,1{1,,1,,2◊2}1{1,,1,,2◊2}2◊2} has recursion level C(C(Ω22+C(Ω2+C(Ω22,σ)2,σ),0),0)
  • {1,,2,,2◊2} has recursion level C(C(Ω22+C(Ω2+C(C(Ω2,C(Ω22,σ)),C(Ω22,σ)),σ),0),0)
  • {1,,1{1,,1,,2◊2}2,,2◊2} has recursion level C(C(Ω22+C(Ω2+C(C(Ω2+C(Ω22,σ),C(Ω22,σ)),C(Ω22,σ)),σ),0),0)
  • {1,,1{1{1,,2,,2◊2}2,,1,,2◊2}2,,2◊2} has recursion level C(C(Ω22+C(Ω2+C(C(Ω2+C(C(Ω2,C(Ω22,σ)),C(Ω22,σ)),C(Ω22,σ)),C(Ω22,σ)),σ),0),0)
  • {1,,1{1,,2,,2◊2}2,,2◊2} has recursion level C(C(Ω22+C(Ω2+C(C(Ω22,C(Ω22,σ)),C(Ω22,σ)),σ),0),0)
  • {1,,1{1,,2,,2◊2}1,2,,2◊2} has recursion level C(C(Ω22+C(Ω2+C(C(Ω22+1,0),C(Ω22,σ)),σ),0),0)
  • {1,,1{1,,2,,2◊2}1{1◊2}2,,2◊2} has recursion level C(C(Ω22+C(Ω2+C(C(Ω22+σ,0),C(Ω22,σ)),σ),0),0)
  • {1,,1{1,,2,,2◊2}1{1,,1,,2◊2}2,,2◊2} has recursion level C(C(Ω22+C(Ω2+C(Ω2,C(Ω22,σ)),σ),0),0)
  • {1,,1,,1,2◊2} has recursion level C(C(Ω22+C(Ω2+C(Ω2,C(Ω22,σ))+1,σ),0),0)
  • {1,,1,,1,,2◊2} has recursion level C(C(Ω22+C(Ω2+C(Ω2,C(Ω22,σ))+C(Ω22,σ),σ),0),0)
  • {1,,1,,1,,1,,2◊2} has recursion level C(C(Ω22+C(Ω2+C(Ω2,C(Ω22,σ))2+C(Ω22,σ),σ),0),0)
  • {1{2,,}2◊2} has recursion level C(C(Ω22+C(Ω2+ω^(C(Ω2,C(Ω22,σ))+1),σ),0),0)
  • {1{1`,,2,,}2◊2} has recursion level C(C(Ω22+C(Ω2+C(C(Ω2,C(Ω2,C(Ω22,σ))),C(Ω2,C(Ω22,σ))),σ),0),0)
  • {1{1,,`1,,2,,}2◊2} has recursion level C(C(Ω22+C(Ω2+C(C(Ω2+C(Ω22,σ),C(Ω22,σ)),C(Ω2,C(Ω22,σ))),σ),0),0)
  • {1{1,,`1`,,2,,}2◊2} has recursion level C(C(Ω22+C(Ω2+C(C(Ω22,C(Ω22,σ)),C(Ω2,C(Ω22,σ))),σ),0),0)
  • {1{1,,`1` `,,2,,}2◊2} has recursion level C(C(Ω22+C(Ω2+C(C(Ω22,C(Ω22,σ)),C(Ω2,C(Ω2,C(Ω22,σ)))),σ),0),0)
  • {1{1,,`1{1,,`1,2,,}2,,}2◊2} has recursion level C(C(Ω22+C(Ω2+C(Ω2+1,C(Ω22,σ)),σ),0),0)
  • {1{1,,`1{1,,`1,,2,,}2,,}2◊2} has recursion level C(C(Ω22+C(Ω2+C(Ω2+C(Ω22,σ),C(Ω22,σ)),σ),0),0)
  • {1{1,,`1{1,,`1`,,2,,}2,,}2◊2} has recursion level C(C(Ω22+C(Ω2+C(Ω2+C(C(Ω22,C(Ω22,σ)),C(Ω2,C(Ω22,σ))),C(Ω22,σ)),σ),0),0)
  • {1{1,,`1{1,,`1{1,,`1,2,,}2,,}2,,}2◊2} has recursion level C(C(Ω22+C(Ω2+C(Ω2+C(Ω2+1,C(Ω22,σ)),C(Ω22,σ)),σ),0),0)
  • {1{1,,`1{1,,`1,,`2,,}2,,}2◊2} has recursion level C(C(Ω22+C(Ω2+C(Ω22,C(Ω22,σ)),σ),0),0)
  • {1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}2◊2} has recursion level C(C(Ω22+C(Ω2+C(Ω22,C(Ω22,σ))+C(Ω2,C(Ω2,C(Ω22,σ))),σ),0),0)
  • {1{1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}1,2,,}2◊2} has recursion level C(C(Ω22+C(Ω2+C(Ω22,C(Ω22,σ))+C(Ω2+1,C(Ω22,σ)),σ),0),0)
  • {1{1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}1`,,2,,}2◊2} has recursion level C(C(Ω22+C(Ω2+C(Ω22,C(Ω22,σ))+C(Ω2+C(Ω22,C(Ω22,σ)),C(Ω22,σ)),σ),0),0)
  • {1{1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}1`,,1`,,2,,}2◊2} has recursion level C(C(Ω22+C(Ω2+C(Ω22,C(Ω22,σ))2,σ),0),0)
  • {1{1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}1` `,,1` `,,2`,,}2◊2} has recursion level C(C(Ω22+C(Ω2+C(Ω22,C(Ω22,C(Ω22,σ)))2,σ),0),0)
  • {1{1{1,,`1,2,,}2,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}2◊2} has recursion level C(C(Ω22+C(Ω2+C(Ω22+1,0),σ),0),0)
  • {1{1{1,,`1{1◊2}2,,}2,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}2◊2} has recursion level C(C(Ω22+C(Ω2+C(Ω22+σ,0),σ),0),0)
  • {1{1{1,,`1{1,,2◊2}2,,}2,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}2◊2} has recursion level C(C(Ω22+C(Ω2+C(Ω22+C(Ω2,σ),0),σ),0),0)
  • {1{1{1,,`1 {1{1{1,,`1,2,,}2,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}2◊2} 2,,}2,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}2◊2} has recursion level C(C(Ω22+C(Ω2+C(Ω22+C(Ω2+C(Ω22+1,0),σ),0),σ),0),0)
  • {1◊3} has recursion level C(C(Ω22+C(Ω22,σ),0),0)
  • {1◊4} has recursion level C(C(Ω22+C(Ω22,C(Ω22,σ)),0),0)
  • {1◊1,2} has recursion level C(C(Ω22+C(Ω22+1,0),0),0)
  • {1◊1{1◊2}2} has recursion level C(C(Ω22+C(Ω22+σ,0),0),0)
  • {1◊1{1◊1,2}2} has recursion level C(C(Ω22+C(Ω22+C(Ω22+1,0),0),0),0)
  • {1◊1,,2} has recursion level C(C(Ω23,0),0)

So {1{1{1,,`1,,2,,}2,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}1,,2} has recursion level C(C(Ω23,0),0).

Here’re some approximations, to make the comparisons above more simple:

  • {1◊2} (where ◊ = {1{1,,`1,,2,,}2,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}) approximately corresponds to C(Ω22,0)
  • {1{1{1,,`1{1◊2}2,,}2,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}2} approximately corresponds to C(Ω2+C(Ω22+C(Ω22,0),0),0)
  • {1,,2◊2} approximately corresponds to C(Ω2,C(Ω22,0))
  • {1{1{1,,`1{1,,2◊2}2,,}2,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}2} approximately corresponds to C(Ω2+C(Ω22+C(Ω2,C(Ω22,0)),0),0)
  • {1,,1{1◊2}2◊2} approximately corresponds to C(Ω2+C(Ω22,0),C(Ω22,0))
  • {1,,1{1,,2◊2}2◊2} approximately corresponds to C(Ω2+C(Ω2,C(Ω22,0)),C(Ω22,0))
  • {1,,1{1,,1,,2◊2}2◊2} approximately corresponds to C(Ω2+C(Ω22,C(Ω22,0)),C(Ω22,0))
  • {1,,1{1,,1,,2◊2}1,2◊2} approximately corresponds to C(Ω2+C(Ω22,C(Ω22,0))+1,C(Ω22,0))
  • {1,,1,,2◊2} approximately corresponds to C(Ω2+C(Ω2,C(Ω22,C(Ω22,0))),C(Ω22,0))
  • {1{1,,`1{1,,`1,2,,}2,,}2◊2} approximately corresponds to C(Ω2+C(Ω2+1,C(Ω22,C(Ω22,0))),C(Ω22,0))
  • {1{1,,`1{1,,`1,,`2,,}2,,}2◊2} approximately corresponds to C(Ω2+C(Ω22,C(Ω22,C(Ω22,0))),C(Ω22,0))
  • {1◊3} approximately corresponds to C(Ω22,C(Ω22,0))
  • {1◊1,2} approximately corresponds to C(Ω22+1,0)
  • {1{1{1,,`1{1◊1,2}2,,}2,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}2} approximately corresponds to C(Ω2+C(Ω22+C(Ω22+1,0),0),0)
  • {1{1{1,,`1{1◊1{1◊2}2}2,,}2,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}2} approximately corresponds to C(Ω2+C(Ω22+C(Ω22+C(Ω22,0),0),0),0)

pDAN vs. Taranovsky’s ordinal notation

Update: under current definition of pDAN, recursion level platforms do not exist.

Here’re the comparisons between pDAN and Taranovsky’s ordinal notation. Due to its high complexity, a separate page is needed. The fundamental sequences of Taranovsky’s notation can be easily defined. Let L(α) be the amount of C’s in standard representation of α, then {\alpha[n]=\max\{\beta|\beta<\alpha\land L(\beta)\le L(\alpha)+n\}}.

Just warm up

  • {1,,1{1,,1,,2}2} has recursion level C(C(Ω22,0),0)
  • {1,,2{1,,1,,2}2} has recursion level C(C(Ω2,C(Ω22,0)),0)
  • {1,,3{1,,1,,2}2} has recursion level C(C(Ω2,C(Ω2,C(Ω22,0))),0)
  • {1,,1,2{1,,1,,2}2} has recursion level C(C(Ω2+1,C(Ω22,0)),0)
  • {1,,1{1,,2}2{1,,1,,2}2} has recursion level C(C(Ω21,C(Ω22,0)),0)
  • {1,,1{1,,3}2{1,,1,,2}2} has recursion level C(C(Ω2+C(Ω21),C(Ω22,0)),0)
  • {1,,1{1,,4}2{1,,1,,2}2} has recursion level C(C(Ω2+C(Ω2,C(Ω21)),C(Ω22,0)),0)
  • {1,,1{1,,1,2}2{1,,1,,2}2} has recursion level C(C(Ω2+C(Ω2+1,0),C(Ω22,0)),0)
  • {1,,1{1,,1,3}2{1,,1,,2}2} has recursion level C(C(Ω2+C(Ω2+2,0),C(Ω22,0)),0)
  • {1,,1{1,,1{1,,2}2}2{1,,1,,2}2} has recursion level C(C(Ω2+C(Ω21,0),C(Ω22,0)),0)
  • {1,,1{1,,1{1,,1,2}2}2{1,,1,,2}2} has recursion level C(C(Ω2+C(Ω2+C(Ω2+1,0),0),C(Ω22,0)),0)
  • {1,,1{1,,1{1,,1{1,,1,2}2}2}2{1,,1,,2}2} has recursion level C(C(Ω2+C(Ω2+C(Ω2+C(Ω2+1,0),0),0),C(Ω22,0)),0)
  • {1,,1{1,,1{1,,1,,2}2}2{1,,1,,2}2} has recursion level C(C(Ω22,C(Ω22,0)),0) (note that C(C(Ω2+C(Ω22,0),C(Ω22,0)),0) is not a valid term)
  • {1,,2{1,,1{1,,1,,2}2}2{1,,1,,2}2} has recursion level C(C(Ω2,C(Ω22,C(Ω22,0))),0)
  • {1,,1{1,,1{1,,1,,2}2}3{1,,1,,2}2} has recursion level C(C(Ω22,C(Ω22,C(Ω22,0))),0)
  • {1,,1{1,,1{1,,1,,2}2}4{1,,1,,2}2} has recursion level C(C(Ω22,C(Ω22,C(Ω22,C(Ω22,0)))),0)
  • {1,,1{1,,1{1,,1,,2}2}1,2{1,,1,,2}2} has recursion level C(C(Ω22+1,0),0) (note that if α is standard, and it doesn’t have subterms over Ω22, then C(C(Ω22+α,0),0) is valid)
  • {1,,1{1,,1{1,,1,,2}2}1{1,,2}2{1,,1,,2}2} has recursion level C(C(Ω22+Ω1,0),0)
  • {1,,1{1,,1{1,,1,,2}2}1{1,,1,2}2{1,,1,,2}2} has recursion level C(C(Ω22+C(Ω2+1,0),0),0)
  • {1,,1{1,,1{1,,1,,2}2}1{1,,1{1,,1,2}2}2{1,,1,,2}2} has recursion level C(C(Ω22+C(Ω2+C(Ω2+1,0),0),0),0)
  • {1,,1{1,,1{1,,1,,2}2}1{1,,1{1,,1,,2}2}2{1,,1,,2}2} has recursion level C(C(Ω22+C(Ω2+C(Ω22,0),0),0),0)

Up to {1,,1{1,,1,,2}3}

Now let separator ■ = {1,,1,,2} and let ordinal a = C(Ω2+C(Ω22,0),0).

  • {1,,1{1,,1■2}2{1,,1■2}2■2} has recursion level C(C(Ω22,C(Ω22+a,0)),0)
  • {1,,1{1,,1■2}1{1,,1■2}3■2} has recursion level C(C(Ω22+a,C(Ω22+a,0)),0)
  • {1,,1{1,,1■2}1{1,,1■2}1,2■2} has recursion level C(C(Ω22+a+1,0),0)
  • {1,,1{1,,1■2}1{1,,1■2}1{1,,1■2}2■2} has recursion level C(C(Ω22+a2,0),0)
  • {1,,1{2,,1■2}2■2} has recursion level C(C(Ω22+ω^(a+1),0),0)
  • {1,,1{1{1,,1■2}2,,1■2}2■2} has recursion level C(C(Ω22+ω^(a2),0),0)
  • {1,,1{1{1,,1■2}1{1,,1■2}2,,1■2}2■2} has recursion level C(C(Ω22+ω^ω^(a2),0),0)
  • {1,,1{1{1{1,,1■2}2,,1■2}2,,1■2}2■2} has recursion level C(C(Ω22+ω^ω^ω^(a2),0),0)
  • {1,,1{1{1,,2■2}2,,1■2}2■2} has recursion level C(C(Ω22+εa+1,0),0) = C(C(Ω22+C(C(Ω2,a),a),0),0)
  • {1,,1{1{1,,2■2}3,,1■2}2■2} has recursion level C(C(Ω22+εa+2,0),0) = C(C(Ω22+C(C(Ω2,a),C(C(Ω2,a),a)),0),0)
  • {1,,1{1{1,,2■2}1,2,,1■2}2■2} has recursion level C(C(Ω22+C(C(Ω2,a)+1,a),0),0)
  • {1,,1{1{1,,2■2}1{1,,2■2}2,,1■2}2■2} has recursion level C(C(Ω22+C(C(Ω2,a)2,a),0),0)
  • {1,,1{1{2,,2■2}2,,1■2}2■2} has recursion level C(C(Ω22+C(ω^(C(Ω2,a)+1),a),0),0)
  • {1,,1{1{1{1,,2■2}2,,2■2}2,,1■2}2■2} has recursion level C(C(Ω22+C(ω^(C(Ω2,a)2),a),0),0)
  • {1,,1{1{1,,3■2}2,,1■2}2■2} has recursion level C(C(Ω22+C(C(Ω2,C(Ω2,a)),a),0),0)
  • {1,,1{1{1,,1,2■2}2,,1■2}2■2} has recursion level C(C(Ω22+C(C(Ω2+1,a),a),0),0)
  • {1,,1{1{1,,1{1,,1■2}2■2}2,,1■2}2■2} has recursion level C(C(Ω22+C(C(Ω2+a,a),a),0),0)
  • {1,,1{1{1,,1{1{1,,1{1,,1■2}2■2}2,,1■2}2■2}2,,1■2}2■2} has recursion level C(C(Ω22+C(C(Ω2+C(C(Ω2+a,a),a),a),a),0),0)
  • {1,,1{1{1,,1{1,,2■2}2■2}2,,1■2}2■2} has recursion level C(C(Ω22+C(C(Ω2+C(Ω2,a),a),a),0),0)

So {1,,1{1,,2■2}2■2} has recursion level C(C(Ω22+C(C(Ω2+C(Ω2,a),a),a),0),0). (The recursion levels of {1,,1{1,,2■2}2■2}, {1{1,,1{1,,2■2}2■2}2,,1■2}, {1,,1{1{1,,1{1,,2■2}2■2}2,,1■2}2■2}, {1{1,,1{1{1,,1{1,,2■2}2■2}2,,1■2}2■2}2,,1■2}, etc. are equal. Separator {1,,1 A 2■2} (where A is any separator) reduces to an expression containing {1,,2■2}, then the {1,,2■2} acquires a layer of { ____ ,,1■2}.)

  • {1,,1{1,,1■2}2{1{1,,1{1,,2■2}2■2}2,,1■2}2■2} has recursion level C(C(Ω22,C(Ω22+C(C(Ω2+C(Ω2,a),a),a),0)),0)
  • {1,,1{1,,1■2}1{1,,1■2}2{1{1,,1{1,,2■2}2■2}2,,1■2}2■2} has recursion level C(C(Ω22+a,C(Ω22+C(C(Ω2+C(Ω2,a),a),a),0)),0)
  • {1,,1{1{1,,2■2}2,,1■2}2{1{1,,1{1,,2■2}2■2}2,,1■2}2■2} has recursion level C(C(Ω22+C(C(Ω2,a),a),C(Ω22+C(C(Ω2+C(Ω2,a),a),a),0)),0)
  • {1,,1{1{1,,1{1,,1■2}2■2}2,,1■2}2{1{1,,1{1,,2■2}2■2}2,,1■2}2■2} has recursion level C(C(Ω22+C(C(Ω2+a,a),a),C(Ω22+C(C(Ω2+C(Ω2,a),a),a),0)),0)
  • {1,,1{1{1,,1{1,,2■2}2■2}2,,1■2}3■2} has recursion level C(C(Ω22+C(C(Ω2+C(Ω2,a),a),a),C(Ω22+C(C(Ω2+C(Ω2,a),a),a),0)),0)
  • {1,,1{1{1,,1{1,,2■2}2■2}2,,1■2}1,2■2} has recursion level C(C(Ω22+C(C(Ω2+C(Ω2,a),a),a)+1,0),0)
  • {1,,1{2{1,,1{1,,2■2}2■2}2,,1■2}2■2} has recursion level C(C(Ω22+ω^(C(C(Ω2+C(Ω2,a),a),a)+1),0),0)
  • {1,,1{1{1,,2■2}2{1,,1{1,,2■2}2■2}2,,1■2}2■2} has recursion level C(C(Ω22+C(C(Ω2,a),C(C(Ω2+C(Ω2,a),a),a)),0),0)
  • {1,,1{1{1,,1,2■2}2{1,,1{1,,2■2}2■2}2,,1■2}2■2} has recursion level C(C(Ω22+C(C(Ω2+1,a),C(C(Ω2+C(Ω2,a),a),a)),0),0)
  • {1,,1{1{1,,1{1{1,,1{1,,2■2}2■2}2,,1■2}2■2}2{1,,1{1,,2■2}2■2}2,,1■2}2■2} has recursion level C(C(Ω22+C(C(Ω2+C(C(Ω2+C(Ω2,a),a),a),a),C(C(Ω2+C(Ω2,a),a),a)),0),0)
  • {1,,1{1{1,,1{1,,2■2}2■2}3,,1■2}2■2} has recursion level C(C(Ω22+C(C(Ω2+C(Ω2,a),a),C(C(Ω2+C(Ω2,a),a),a)),0),0)
  • {1,,1{1{1,,1{1,,2■2}2■2}1,2,,1■2}2■2} has recursion level C(C(Ω22+C(C(Ω2+C(Ω2,a),a)+1,a),0),0)
  • {1,,1{1{1,,1{1,,2■2}2■2}1{1,,1■2}2,,1■2}2■2} has recursion level C(C(Ω22+C(C(Ω2+C(Ω2,a),a)+a,a),0),0)
  • {1,,1{1{1,,1{1,,2■2}2■2}1{1{1,,1{1,,2■2}2■2}2,,1■2}2,,1■2}2■2} has recursion level C(C(Ω22+C(C(Ω2+C(Ω2,a),a)+C(C(Ω2+C(Ω2,a),a),a),a),0),0)
  • {1,,1{1{1,,1{1,,2■2}2■2}1{1,,2■2}2,,1■2}2■2} has recursion level C(C(Ω22+C(C(Ω2+C(Ω2,a),a)+C(Ω2,a),a),0),0)
  • {1,,1{1{1,,1{1,,2■2}2■2}1{1{1,,3■2}2,,2■2}2,,1■2}2■2} has recursion level C(C(Ω22+C(C(Ω2+C(Ω2,a),a)+C(C(Ω2,C(Ω2,a)),C(Ω2,a)),a),0),0)
  • {1,,1{1{1,,1{1,,2■2}2■2}1{1,,3■2}2,,1■2}2■2} has recursion level C(C(Ω22+C(C(Ω2+C(Ω2,a),a)+C(C(Ω2,C(Ω2,a)),C(Ω2,a)),a),0),0)
  • {1,,1{1{1,,1{1,,2■2}2■2}1{1{1,,4■2}2,,3■2}2,,1■2}2■2} has recursion level C(C(Ω22+C(C(Ω2+C(Ω2,a),a)+C(C(Ω2,C(Ω2,C(Ω2,a))),C(Ω2,a)),a),0),0)
  • {1,,1{1{1,,1{1,,2■2}2■2}1{1,,4■2}2,,1■2}2■2} has recursion level C(C(Ω22+C(C(Ω2+C(Ω2,a),a)+C(C(Ω2,C(Ω2,C(Ω2,a))),C(Ω2,a)),a),0),0) (when solving this term, by the adding rule, the {1,,4■2} will reduce to {1{1,,4■2}2,,3■2}, then {1{1{1,,4■2}2,,3■2}2,,2■2})
  • {1,,1{1{1,,1{1,,2■2}2■2}1{1,,1,2■2}2,,1■2}2■2} has recursion level C(C(Ω22+C(C(Ω2+C(Ω2,a),a)+C(C(Ω2+1,a),C(Ω2,a)),a),0),0)
  • {1,,1{1{1,,1{1,,2■2}2■2}1{1,,1{1,,1■2}2■2}2,,1■2}2■2} has recursion level C(C(Ω22+C(C(Ω2+C(Ω2,a),a)+C(C(Ω2+a,a),C(Ω2,a)),a),0),0)
  • {1,,1{1{1,,1{1,,2■2}2■2}1{1,,1{1{1,,1{1,,1■2}2■2}2,,1■2}2■2}2,,1■2}2■2} has recursion level C(C(Ω22+C(C(Ω2+C(Ω2,a),a)+C(C(Ω2+C(C(Ω2+a,a),a),a),C(Ω2,a)),a),0),0)
  • {1,,1{1{1,,1{1,,2■2}2■2}1{1,,1{1{1,,1{1,,2■2}2■2}2,,1■2}2■2}2,,1■2}2■2} has recursion level C(C(Ω22+C(C(Ω2+C(Ω2,a),a)+C(C(Ω2+C(C(Ω2+C(Ω2,a),a),a),a),C(Ω2,a)),a),0),0)
  • {1,,1{1{1,,1{1,,2■2}2■2}1{1,,1{1,,2■2}2■2}2,,1■2}2■2} has recursion level C(C(Ω22+C(C(Ω2+C(Ω2,a),a)+C(C(Ω2+C(Ω2,a),a),C(Ω2,a)),a),0),0)
  • {1,,1{1{1,,1{1,,2■2}2■2}1{1,,1{1,,2■2}2■2}1,2,,1■2}2■2} has recursion level C(C(Ω22+C(C(Ω2+C(Ω2,a),a)+C(C(Ω2+C(Ω2,a),a),C(Ω2,a))+1,a),0),0)
  • {1,,1{1{1,,1{1,,2■2}2■2}1{1,,1{1,,2■2}2■2}1{1,,1{1,,2■2}2■2}2,,1■2}2■2} has recursion level C(C(Ω22+C(C(Ω2+C(Ω2,a),a)+C(C(Ω2+C(Ω2,a),a),C(Ω2,a))2,a),0),0)
  • {1,,1{1{2,,1{1,,2■2}2■2}2,,1■2}2■2} has recursion level C(C(Ω22+C(C(Ω2+C(Ω2,a),a)+ω^(C(C(Ω2+C(Ω2,a),a),C(Ω2,a))+1),a),0),0)
  • {1,,1{1{1{1,,1■2}2,,1{1,,2■2}2■2}2,,1■2}2■2} has recursion level C(C(Ω22+C(C(Ω2+C(Ω2,a),a)+ω^(C(C(Ω2+C(Ω2,a),a),C(Ω2,a))+a),a),0),0)
  • {1,,1{1{1{1,,2■2}2,,1{1,,2■2}2■2}2,,1■2}2■2} has recursion level C(C(Ω22+C(C(Ω2+C(Ω2,a),a)+ω^(C(C(Ω2+C(Ω2,a),a),C(Ω2,a))+C(Ω2,a)),a),0),0)

So {1{1,,2■2}2,,1{1,,2■2}2■2} has recursion level C(C(Ω22+C(C(Ω2+C(Ω2,a),a)+ω^(C(C(Ω2+C(Ω2,a),a),C(Ω2,a))+C(Ω2,a)),a),0),0). (the recursion levels of {1{1{1,,2■2}2,,1{1,,2■2}2■2}2,,1■2}, {1,,1{1{1{1,,2■2}2,,1{1,,2■2}2■2}2,,1■2}2■2} and {1{1,,1{1{1{1,,2■2}2,,1{1,,2■2}2■2}2,,1■2}2■2}2,,1■2} are equal)

  • {1,,1{1{1{1,,1{1,,2■2}2■2}2,,2■2}2,,1■2}2■2} has recursion level C(C(Ω22+C(C(Ω2+C(Ω2,a),a),a),0),0)
  • {1,,1{1{1,,2■2}2{1{1,,1{1,,2■2}2■2}2,,2■2}2,,1■2}2■2} has recursion level C(C(Ω22+C(C(Ω2,a),C(C(Ω2+C(Ω2,a),a),a)),0),0)
  • {1,,1{1{1{1,,1{1,,2■2}2■2}2,,2■2}3,,1■2}2■2} has recursion level C(C(Ω22+C(C(Ω2+C(Ω2,a),a),C(C(Ω2+C(Ω2,a),a),a)),0),0)
  • {1,,1{1{1{1,,1{1,,2■2}2■2}2,,2■2}1,2,,1■2}2■2} has recursion level C(C(Ω22+C(C(Ω2+C(Ω2,a),a)+1,a),0),0)
  • {1,,1{1{1{1,,1{1,,2■2}2■2}2,,2■2}1{1,,2■2}2,,1■2}2■2} has recursion level C(C(Ω22+C(C(Ω2+C(Ω2,a),a)+C(Ω2,a),a),0),0)
  • {1,,1{1{1{1,,1{1,,2■2}2■2}2,,2■2}1{1{1,,1{1,,1■2}2■2}2,,2■2}2,,1■2}2■2} has recursion level C(C(Ω22+C(C(Ω2+C(Ω2,a),a)+C(C(Ω2+a,a),C(Ω2,a)),a),0),0)
  • {1,,1{1{1{1,,1{1,,2■2}2■2}2,,2■2}1{1{1,,1{1,,2■2}2■2}2,,2■2}2,,1■2}2■2} has recursion level C(C(Ω22+C(C(Ω2+C(Ω2,a),a)+C(C(Ω2+C(Ω2,a),a),C(Ω2,a)),a),0),0)
  • {1,,1{1{2{1,,1{1,,2■2}2■2}2,,2■2}2,,1■2}2■2} has recursion level C(C(Ω22+C(C(Ω2+C(Ω2,a),a)+ω^(C(C(Ω2+C(Ω2,a),a),C(Ω2,a))+1),a),0),0)
  • {1,,1{1{1{1,,3■2}2{1,,1{1,,2■2}2■2}2,,2■2}2,,1■2}2■2} has recursion level C(C(Ω22+C(C(Ω2+C(Ω2,a),a)+C(C(Ω2,C(Ω2,a)),C(C(Ω2+C(Ω2,a),a),C(Ω2,a))),a),0),0)
  • {1,,1{1{1{1,,1{1,,1■2}2■2}2{1,,1{1,,2■2}2■2}2,,2■2}2,,1■2}2■2} has recursion level C(C(Ω22+C(C(Ω2+C(Ω2,a),a)+C(C(Ω2+a,a),C(C(Ω2+C(Ω2,a),a),C(Ω2,a))),a),0),0)
  • {1,,1{1{1{1,,1{1,,2■2}2■2}3,,2■2}2,,1■2}2■2} has recursion level C(C(Ω22+C(C(Ω2+C(Ω2,a),a)+C(C(Ω2+C(Ω2,a),a),C(C(Ω2+C(Ω2,a),a),C(Ω2,a))),a),0),0)
  • {1,,1{1{1{1,,1{1,,2■2}2■2}1,2,,2■2}2,,1■2}2■2} has recursion level C(C(Ω22+C(C(Ω2+C(Ω2,a),a)+C(C(Ω2+C(Ω2,a),a)+1,C(Ω2,a)),a),0),0)
  • {1,,1{1{1{1,,1{1,,2■2}2■2}1{1,,2■2}2,,2■2}2,,1■2}2■2} has recursion level C(C(Ω22+C(C(Ω2+C(Ω2,a),a)+C(C(Ω2+C(Ω2,a),a)+C(Ω2,a),C(Ω2,a)),a),0),0)
  • {1,,1{1{1{1,,1{1,,2■2}2■2}1{1,,3■2}2,,2■2}2,,1■2}2■2} has recursion level C(C(Ω22+C(C(Ω2+C(Ω2,a),a)+C(Ω2,C(Ω2,a)),a),0),0)
  • {1,,1{1{1{1,,1{1,,2■2}2■2}2,,2■2}2{1{1,,1{1,,2■2}2■2}1{1,,3■2}2,,2■2}2,,1■2}2■2} has recursion level C(C(Ω22+C(C(Ω2+C(Ω2,a),a),C(C(Ω2+C(Ω2,a),a)+C(Ω2,C(Ω2,a)),a)),0),0)
  • {1,,1{1{1{1,,1{1,,2■2}2■2}2,,2■2}1{1{1,,1{1,,1■2}2■2}2,,2■2}2 {1{1,,1{1,,2■2}2■2}1{1,,3■2}2,,2■2}2,,1■2}2■2} has recursion level C(C(Ω22+C(C(Ω2+C(Ω2,a),a)+C(C(Ω2+a,a),C(Ω2,a)),C(C(Ω2+C(Ω2,a),a)+C(Ω2,C(Ω2,a)),a)),0),0)
  • {1,,1{1{1{1,,1{1,,2■2}2■2}2,,2■2}1{1{1,,1{1 A 2,,1■2}2■2}2,,2■2}2 A 2,,1■2}2■2} (where A = {1{1,,1{1,,2■2}2■2}1{1,,3■2}2,,2■2}) has recursion level C(C(Ω22+C(C(Ω2+C(Ω2,a),a)+C(C(Ω2+C(C(Ω2+C(Ω2,a),a)+C(Ω2,C(Ω2,a)),a),a),C(Ω2,a)),C(C(Ω2+C(Ω2,a),a)+C(Ω2,C(Ω2,a)),a)),0),0)
  • {1,,1{1{1{1,,1{1,,2■2}2■2}2,,2■2}1{1{1,,1{1,,2■2}2■2}2,,2■2}2 A 2,,1■2}2■2} (where A = {1{1,,1{1,,2■2}2■2}1{1,,3■2}2,,2■2}) has recursion level C(C(Ω22+C(C(Ω2+C(Ω2,a),a)+C(C(Ω2+C(Ω2,a),a),C(Ω2,a)),C(C(Ω2+C(Ω2,a),a)+C(Ω2,C(Ω2,a)),a)),0),0)
  • {1,,1{1{1{1,,1{1,,2■2}2■2}1,2,,2■2}2{1{1,,1{1,,2■2}2■2}1{1,,3■2}2,,2■2}2,,1■2}2■2} has recursion level C(C(Ω22+C(C(Ω2+C(Ω2,a),a)+C(C(Ω2+C(Ω2,a),a)+1,C(Ω2,a)),C(C(Ω2+C(Ω2,a),a)+C(Ω2,C(Ω2,a)),a)),0),0)
  • {1,,1{1{1{1,,1{1,,2■2}2■2}1{1,,2■2}2,,2■2}2{1{1,,1{1,,2■2}2■2}1{1,,3■2}2,,2■2}2,,1■2}2■2} has recursion level C(C(Ω22+C(C(Ω2+C(Ω2,a),a)+C(C(Ω2+C(Ω2,a),a)+C(Ω2,a),C(Ω2,a)),C(C(Ω2+C(Ω2,a),a)+C(Ω2,C(Ω2,a)),a)),0),0)
  • {1,,1{1{1{1,,1{1,,2■2}2■2}1{1,,3■2}2,,2■2}3,,1■2}2■2} has recursion level C(C(Ω22+C(C(Ω2+C(Ω2,a),a)+C(Ω2,C(Ω2,a)),C(C(Ω2+C(Ω2,a),a)+C(Ω2,C(Ω2,a)),a)),0),0)
  • {1,,1{1{1{1,,1{1,,2■2}2■2}1{1,,3■2}2,,2■2}1,2,,1■2}2■2} has recursion level C(C(Ω22+C(C(Ω2+C(Ω2,a),a)+C(Ω2,C(Ω2,a))+1,a),0),0)
  • {1,,1{1{1{1,,1{1,,2■2}2■2}1{1,,3■2}2,,2■2}1{1,,2■2}2,,1■2}2■2} has recursion level C(C(Ω22+C(C(Ω2+C(Ω2,a),a)+C(Ω2,C(Ω2,a))+C(Ω2,a),a),0),0)
  • {1,,1{1{1{1,,1{1,,2■2}2■2}1{1,,3■2}2,,2■2}1{1{1,,1{1,,1■2}2■2}2,,2■2}2,,1■2}2■2} has recursion level C(C(Ω22+C(C(Ω2+C(Ω2,a),a)+C(Ω2,C(Ω2,a))+C(C(Ω2+a,a),C(Ω2,a)),a),0),0)
  • {1,,1{1{1{1,,1{1,,2■2}2■2}1{1,,3■2}2,,2■2}1{1{1,,1{1,,2■2}2■2}2,,2■2}2,,1■2}2■2} has recursion level C(C(Ω22+C(C(Ω2+C(Ω2,a),a)+C(Ω2,C(Ω2,a))+C(C(Ω2+C(Ω2,a),a),C(Ω2,a)),a),0),0)
  • {1,,1{1{1{1,,1{1,,2■2}2■2}1{1,,3■2}2,,2■2}1{1{1,,1{1,,2■2}2■2}1,2,,2■2}2,,1■2}2■2} has recursion level C(C(Ω22+C(C(Ω2+C(Ω2,a),a)+C(Ω2,C(Ω2,a))+C(C(Ω2+C(Ω2,a),a)+1,C(Ω2,a)),a),0),0)
  • {1,,1{1{1{1,,1{1,,2■2}2■2}1{1,,3■2}2,,2■2}1{1{1,,1{1,,2■2}2■2}1{1,,2■2}2,,2■2}2,,1■2}2■2} has recursion level C(C(Ω22+C(C(Ω2+C(Ω2,a),a)+C(Ω2,C(Ω2,a))+C(C(Ω2+C(Ω2,a),a)+C(Ω2,a),C(Ω2,a)),a),0),0)
  • {1,,1{1{1{1,,1{1,,2■2}2■2}1{1,,3■2}2,,2■2}1{1{1,,1{1,,2■2}2■2}1{1,,3■2}2,,2■2}2,,1■2}2■2} has recursion level C(C(Ω22+C(C(Ω2+C(Ω2,a),a)+C(Ω2,C(Ω2,a))+C(C(Ω2+C(Ω2,a),a)+C(Ω2,C(Ω2,a)),C(Ω2,a)),a),0),0)
  • {1,,1{1{1{1,,1{1,,2■2}2■2}1{1,,3■2}2,,2■2}1{1{1,,1{1,,2■2}2■2}1{1,,3■2}2,,2■2}1,2,,1■2}2■2} has recursion level C(C(Ω22+C(C(Ω2+C(Ω2,a),a)+C(Ω2,C(Ω2,a))+C(C(Ω2+C(Ω2,a),a)+C(Ω2,C(Ω2,a)),C(Ω2,a))+1,a),0),0)
  • {1,,1{1{2{1,,1{1,,2■2}2■2}1{1,,3■2}2,,2■2}2,,1■2}2■2} has recursion level C(C(Ω22+C(C(Ω2+C(Ω2,a),a)+C(Ω2,C(Ω2,a))+ω^(C(C(Ω2+C(Ω2,a),a)+C(Ω2,C(Ω2,a)),C(Ω2,a))+1),a),0),0)
  • {1,,1{1{1{1,,3■2}2{1,,1{1,,2■2}2■2}1{1,,3■2}2,,2■2}2,,1■2}2■2} has recursion level C(C(Ω22+C(C(Ω2+C(Ω2,a),a)+C(Ω2,C(Ω2,a))+C(C(Ω2,C(Ω2,a)),C(C(Ω2+C(Ω2,a),a)+C(Ω2,C(Ω2,a)),C(Ω2,a))),a),0),0)
  • {1,,1{1{1{1,,1{1,,2■2}2■2}2{1,,3■2}2,,2■2}2,,1■2}2■2} has recursion level C(C(Ω22+C(C(Ω2+C(Ω2,a),a)+C(Ω2,C(Ω2,a))+C(C(Ω2+C(Ω2,a),a),C(C(Ω2+C(Ω2,a),a)+C(Ω2,C(Ω2,a)),C(Ω2,a))),a),0),0)
  • {1,,1{1{1{1,,1{1,,2■2}2■2}1{1,,2■2}2{1,,3■2}2,,2■2}2,,1■2}2■2} has recursion level C(C(Ω22+C(C(Ω2+C(Ω2,a),a)+C(Ω2,C(Ω2,a))+C(C(Ω2+C(Ω2,a),a)+C(Ω2,a),C(C(Ω2+C(Ω2,a),a)+C(Ω2,C(Ω2,a)),C(Ω2,a))),a),0),0)
  • {1,,1{1{1{1,,1{1,,2■2}2■2}1{1,,3■2}3,,2■2}2,,1■2}2■2} has recursion level C(C(Ω22+C(C(Ω2+C(Ω2,a),a)+C(Ω2,C(Ω2,a))+C(C(Ω2+C(Ω2,a),a)+C(Ω2,C(Ω2,a)),C(C(Ω2+C(Ω2,a),a)+C(Ω2,C(Ω2,a)),C(Ω2,a))),a),0),0)
  • {1,,1{1{1{1,,1{1,,2■2}2■2}1{1,,3■2}1,2,,2■2}2,,1■2}2■2} has recursion level C(C(Ω22+C(C(Ω2+C(Ω2,a),a)+C(Ω2,C(Ω2,a))+C(C(Ω2+C(Ω2,a),a)+C(Ω2,C(Ω2,a))+1,C(Ω2,a)),a),0),0)
  • {1,,1{1{1{1,,1{1,,2■2}2■2}1{1,,3■2}1{1{1,,1{1,,2■2}2■2}1{1,,3■2}1,2,,2■2}2,,2■2}2,,1■2}2■2} has recursion level C(C(Ω22+C(C(Ω2+C(Ω2,a),a)+C(Ω2,C(Ω2,a))+C(C(Ω2+C(Ω2,a),a)+C(Ω2,C(Ω2,a))+C(C(Ω2+C(Ω2,a),a)+C(Ω2,C(Ω2,a))+1,C(Ω2,a)),C(Ω2,a)),a),0),0)
  • {1,,1{1{1{1,,1{1,,2■2}2■2}1{1,,3■2}1{1,,3■2}2,,2■2}2,,1■2}2■2} has recursion level C(C(Ω22+C(C(Ω2+C(Ω2,a),a)+C(Ω2,C(Ω2,a))2,a),0),0)
  • {1,,1{1{1{1,,1{1,,2■2}2■2}1{2,,3■2}2,,2■2}2,,1■2}2■2} has recursion level C(C(Ω22+C(C(Ω2+C(Ω2,a),a)+ω^(C(Ω2,C(Ω2,a))+1),a),0),0)
  • {1,,1{1{1{1,,1{1,,2■2}2■2}1{1,,4■2}2,,2■2}2,,1■2}2■2} has recursion level C(C(Ω22+C(C(Ω2+C(Ω2,a),a)+C(C(Ω2,C(Ω2,C(Ω2,a))),C(Ω2,C(Ω2,a))),a),0),0)
  • {1,,1{1{1{1,,1{1,,2■2}2■2}1{1,,1,2■2}2,,2■2}2,,1■2}2■2} has recursion level C(C(Ω22+C(C(Ω2+C(Ω2,a),a)+C(C(Ω2+1,a),C(Ω2,C(Ω2,a))),a),0),0)
  • {1,,1{1{1{1,,1{1,,2■2}2■2}1{1,,1{1,,2■2}2■2}2,,2■2}2,,1■2}2■2} has recursion level C(C(Ω22+C(C(Ω2+C(Ω2,a),a)+C(C(Ω2+C(Ω2,a),a),C(Ω2,C(Ω2,a))),a),0),0)
  • {1,,1{1{1{2,,1{1,,2■2}2■2}2,,2■2}2,,1■2}2■2} has recursion level C(C(Ω22+C(C(Ω2+C(Ω2,a),a)+ω^(C(C(Ω2+C(Ω2,a),a),C(Ω2,C(Ω2,a)))+1),a),0),0)
  • {1,,1{1{1{1{1,,2■2}2,,1{1,,2■2}2■2}2,,2■2}2,,1■2}2■2} has recursion level C(C(Ω22+C(C(Ω2+C(Ω2,a),a)+ω^(C(C(Ω2+C(Ω2,a),a),C(Ω2,C(Ω2,a)))+C(Ω2,a)),a),0),0)
  • {1,,1{1{1{1{1,,3■2}2,,1{1,,2■2}2■2}2,,2■2}2,,1■2}2■2} has recursion level C(C(Ω22+C(C(Ω2+C(Ω2,a),a)+ω^(C(C(Ω2+C(Ω2,a),a),C(Ω2,C(Ω2,a)))+C(Ω2,C(Ω2,a))),a),0),0)

So {1{1,,3■2}2,,1{1,,2■2}2■2} has recursion level C(C(Ω22+C(C(Ω2+C(Ω2,a),a)+ω^(C(C(Ω2+C(Ω2,a),a),C(Ω2,C(Ω2,a)))+C(Ω2,C(Ω2,a))),a),0),0). And further, the recursion level of {1{1,,k■2}2,,1{1,,2■2}2■2} is between C(C(Ω22+C(C(Ω2+C(Ω2,a),a)+C(Ω2,C(Ω2,…C(Ω2,a))),a),0),0) (with k-1 Ω2′s in the blue part) and C(C(Ω22+C(C(Ω2+C(Ω2,a),a)+C(Ω2,C(Ω2,…C(Ω2,a))),a),0),0) (with k Ω2′s in the blue part), so {1{1,,1,2■2}2,,1{1,,2■2}2■2} has recursion level C(C(Ω22+C(C(Ω2+C(Ω2,a),a)+C(Ω2+1,a),a),0),0).

  • {1,,1{1{1{1,,1,2■2}2,,1{1,,2■2}2■2}2,,1■2}2■2} has recursion level C(C(Ω22+C(C(Ω2+C(Ω2,a),a)+C(Ω2+1,a),a),0),0)
  • {1,,1{1{1,,1{1,,2■2}2■2}2{1{1,,1,2■2}2,,1{1,,2■2}2■2}2,,1■2}2■2} has recursion level C(C(Ω22+C(C(Ω2+C(Ω2,a),a),C(C(Ω2+C(Ω2,a),a)+C(Ω2+1,a),a)),0),0)
  • {1,,1{1{1{1,,2■2}2,,1{1,,2■2}2■2}2{1{1,,1,2■2}2,,1{1,,2■2}2■2}2,,1■2}2■2} has recursion level C(C(Ω22+C(C(Ω2+C(Ω2,a),a)+ω^(C(C(Ω2+C(Ω2,a),a),C(Ω2,a))+C(Ω2,a)),C(C(Ω2+C(Ω2,a),a)+C(Ω2+1,a),a)),0),0)
  • {1,,1{1{1{1,,3■2}2,,1{1,,2■2}2■2}2{1{1,,1,2■2}2,,1{1,,2■2}2■2}2,,1■2}2■2} has recursion level C(C(Ω22+C(C(Ω2+C(Ω2,a),a)+ω^(C(C(Ω2+C(Ω2,a),a),C(Ω2,C(Ω2,a)))+C(Ω2,C(Ω2,a))),C(C(Ω2+C(Ω2,a),a)+C(Ω2+1,a),a)),0),0)
  • {1,,1{1{1{1,,1,2■2}2,,1{1,,2■2}2■2}3,,1■2}2■2} has recursion level C(C(Ω22+C(C(Ω2+C(Ω2,a),a)+C(Ω2+1,a),C(C(Ω2+C(Ω2,a),a)+C(Ω2+1,a),a)),0),0)
  • {1,,1{1{1{1,,1,2■2}2,,1{1,,2■2}2■2}1,2,,1■2}2■2} has recursion level C(C(Ω22+C(C(Ω2+C(Ω2,a),a)+C(Ω2+1,a)+1,a),0),0)
  • {1{1,,2■2}2{1,,1,2■2}2,,1{1,,2■2}2■2} has recursion level C(C(Ω22+C(C(Ω2+C(Ω2,a),a)+C(Ω2+1,a)+ω^(C(C(Ω2+C(Ω2,a),a)+C(Ω2+1,a),C(Ω2,a))+C(Ω2,a)),a),0),0)
  • {1{1,,3■2}2{1,,1,2■2}2,,1{1,,2■2}2■2} has recursion level C(C(Ω22+C(C(Ω2+C(Ω2,a),a)+C(Ω2+1,a)+ω^(C(C(Ω2+C(Ω2,a),a)+C(Ω2+1,a),C(Ω2,C(Ω2,a)))+C(Ω2,C(Ω2,a))),a),0),0)
  • {1{1,,1,2■2}3,,1{1,,2■2}2■2} has recursion level C(C(Ω22+C(C(Ω2+C(Ω2,a),a)+C(Ω2+1,a)2,a),0),0)
  • {1{1,,2,2■2}2,,1{1,,2■2}2■2} has recursion level C(C(Ω22+C(C(Ω2+C(Ω2,a),a)+C(Ω2,C(Ω2+1,a)),a),0),0)
  • {1{1,,1,3■2}2,,1{1,,2■2}2■2} has recursion level C(C(Ω22+C(C(Ω2+C(Ω2,a),a)+C(Ω2+1,C(Ω2+1,a)),a),0),0)
  • {1{1,,1,1,2■2}2,,1{1,,2■2}2■2} has recursion level C(C(Ω22+C(C(Ω2+C(Ω2,a),a)+C(Ω2+2,a),a),0),0)
  • {1{1,,1{1,,1■2}2■2}2,,1{1,,2■2}2■2} has recursion level C(C(Ω22+C(C(Ω2+C(Ω2,a),a)+C(Ω2+a,a),a),0),0)
  • {1{1,,1{1{1{1,,1,2■2}2,,1{1,,2■2}2■2}2,,1■2}2■2}2,,1{1,,2■2}2■2} has recursion level C(C(Ω22+C(C(Ω2+C(Ω2,a),a)+C(Ω2+C(C(Ω2+C(Ω2,a),a)+C(Ω2+1,a),a),a),a),0),0)
  • {1{1,,1{1,,2■2}2■2}2,,1{1,,2■2}2■2} has recursion level C(C(Ω22+C(C(Ω2+C(Ω2,a),a)2,a),0),0)
  • {1{1,,1,2■2}2{1,,1{1,,2■2}2■2}2,,1{1,,2■2}2■2} has recursion level C(C(Ω22+C(C(Ω2+C(Ω2,a),a)2+C(Ω2+1,a),a),0),0)
  • {1{1,,1{1,,2■2}2■2}3,,1{1,,2■2}2■2} has recursion level C(C(Ω22+C(C(Ω2+C(Ω2,a),a)3,a),0),0)
  • {1{1,,1{1,,2■2}2■2}1,2,,1{1,,2■2}2■2} has recursion level C(C(Ω22+C(ω^(C(Ω2+C(Ω2,a),a)+1),a),0),0)
  • {1{1,,1{1,,2■2}2■2}1{1,,1{1,,2■2}2■2}2,,1{1,,2■2}2■2} has recursion level C(C(Ω22+C(ω^(C(Ω2+C(Ω2,a),a)2),a),0),0)
  • {1{2,,1{1,,2■2}2■2}2,,1{1,,2■2}2■2} has recursion level C(C(Ω22+C(ω^ω^(C(Ω2+C(Ω2,a),a)+1),a),0),0)
  • {1,,2{1,,2■2}2■2} has recursion level C(C(Ω22+C(C(Ω2,C(Ω2+C(Ω2,a),a)),a),0),0)
  • {1,,1{1,,1■2}2{1,,2■2}2■2} has recursion level C(C(Ω22+C(C(Ω2+a,C(Ω2+C(Ω2,a),a)),a),0),0)
  • {1,,1{1,,2■2}3■2} has recursion level C(C(Ω22+C(C(Ω2+C(Ω2,a),C(Ω2+C(Ω2,a),a)),a),0),0)
  • {1,,1{1,,2■2}1,2■2} has recursion level C(C(Ω22+C(C(Ω2+C(Ω2,a)+1,a),a),0),0)
  • {1,,1{1,,2■2}1{1,,2■2}2■2} has recursion level C(C(Ω22+C(C(Ω2+C(Ω2,a)2,a),a),0),0)
  • {1,,1{2,,2■2}2■2} has recursion level C(C(Ω22+C(C(Ω2+ω^(C(Ω2,a)+1),a),a),0),0)
  • {1,,1{1{1,,3■2}2,,2■2}2■2} has recursion level C(C(Ω22+C(C(Ω2+C(C(Ω2,C(Ω2,a)),C(Ω2,a)),a),a),0),0)
  • {1,,1{1{1,,1{1,,2■2}2■2}2,,2■2}2■2} has recursion level C(C(Ω22+C(C(Ω2+C(C(Ω2+C(Ω2,a),a),C(Ω2,a)),a),a),0),0)
  • {1,,1{1{1,,1{1{1,,1{1,,2■2}2■2}2,,2■2}2■2}2,,2■2}2■2} has recursion level C(C(Ω22+C(C(Ω2+C(C(Ω2+C(C(Ω2+C(Ω2,a),a),C(Ω2,a)),a),C(Ω2,a)),a),a),0),0)
  • {1,,1{1,,3■2}2■2} has recursion level C(C(Ω22+C(C(Ω2+C(Ω2,C(Ω2,a)),a),a),0),0)
  • {1,,1{1,,4■2}2■2} has recursion level C(C(Ω22+C(C(Ω2+C(Ω2,C(Ω2,C(Ω2,a))),a),a),0),0)
  • {1,,1{1,,1,2■2}2■2} has recursion level C(C(Ω22+C(C(Ω2+C(Ω2+1,a),a),a),0),0)
  • {1,,1{1,,1{1,,1■2}2■2}2■2} has recursion level C(C(Ω22+C(C(Ω2+C(Ω2+a,a),a),a),0),0)
  • {1,,1{1,,1{1,,2■2}2■2}2■2} has recursion level C(C(Ω22+C(C(Ω2+C(Ω2+C(Ω2,a),a),a),a),0),0)
  • {1,,1{1,,1{1,,1{1,,1■2}2■2}2■2}2■2} has recursion level C(C(Ω22+C(C(Ω2+C(Ω2+C(Ω2+a,a),a),a),a),0),0)
  • {1,,1■3} has recursion level C(C(Ω22+C(C(Ω22,0),a),0),0)

So {1,,1{1,,1,,2}3} has recursion level C(C(Ω22+C(C(Ω22,0),C(Ω2+C(Ω22,0),0)),0),0). Here’re some approximations, to make the comparisons above more simple:

  • {1,,1■2} (where separator ■ = {1,,1,,2}) approximately corresponds to a = C(Ω2+C(Ω22,0),0)
  • {1{1,,2■2}2,,1■2} approximately corresponds to C(C(Ω2,a),a)
  • {1,,2■2} approximately corresponds to C(Ω2,a)
  • {1,,3■2} approximately corresponds to C(Ω2,C(Ω2,a))
  • {1,,1,2■2} approximately corresponds to C(Ω2+1,a)
  • {1,,1{1,,1■2}2■2} approximately corresponds to C(Ω2+a,a)
  • {1{1,,1{1,,1■2}2■2}2,,1■2} approximately corresponds to C(C(Ω2+a,a),a)
  • {1,,1{1,,2■2}2■2} approximately corresponds to C(Ω2+C(Ω2,a),a)
  • {1{1,,1{1,,2■2}2■2}2,,1■2} approximately corresponds to C(C(Ω2+C(Ω2,a),a),a)
  • {1{1,,1{1,,2■2}2■2}3,,1■2} approximately corresponds to C(C(Ω2+C(Ω2,a),a),C(C(Ω2+C(Ω2,a),a),a))
  • {1{1,,1{1,,2■2}2■2}1,2,,1■2} approximately corresponds to C(C(Ω2+C(Ω2,a),a)+1,a)
  • {1,,2{1,,2■2}2■2} approximately corresponds to C(Ω2,C(Ω2+C(Ω2,a),a))
  • {1,,1{1,,2■2}3■2} approximately corresponds to C(Ω2+C(Ω2,a),C(Ω2+C(Ω2,a),a))
  • {1,,1{1,,2■2}1,2■2} approximately corresponds to C(Ω2+C(Ω2,a)+1,a)
  • {1,,1{1,,2■2}1{1,,2■2}2■2} approximately corresponds to C(Ω2+C(Ω2,a)2,a)
  • {1,,1{1,,3■2}2■2} approximately corresponds to C(Ω2+C(Ω2,C(Ω2,a)),a)
  • {1,,1{1,,1,2■2}2■2} approximately corresponds to C(Ω2+C(Ω2+1,a),a)
  • {1,,1{1,,1{1,,1■2}2■2}2■2} approximately corresponds to C(Ω2+C(Ω2+a,a),a)

Up to {1,,2,,2}

Still, let separator ■ = {1,,1,,2} and let ordinal a = C(Ω2+C(Ω22,0),0).

  • {1,,1{1{1,,1■3}2,,1■2}2■2} has recursion level C(C(Ω22+C(C(Ω22,0),a),0),0) (the recursion levels of {1,,1■3}, {1,,1{1,,1■3}2■2}, {1{1,,1■3}2,,1■2}, {1{1,,1{1,,1■3}2■2}2,,1■2}, {1,,1{1{1,,1{1,,1■3}2■2}2,,1■2}2■2}, etc. are equal)
  • {1,,1{1{1,,1■3}2,,1■2}1,2■2} has recursion level C(C(Ω22+C(C(Ω22,0),a)+1,0),0)
  • {1,,1{1{1,,1■3}2,,1■2}1{1,,1■2}2■2} has recursion level C(C(Ω22+C(C(Ω22,0),a)+a,0),0)
  • {1,,1{1{1,,1■3}2,,1■2}1{1{1,,1,2■2}2,,1■2}2■2} has recursion level C(C(Ω22+C(C(Ω22,0),a)+C(C(Ω2+1,a),a),0),0)
  • {1,,1{1{1,,1■3}2,,1■2}1{1{1,,1{1,,1,2■2}2■2}2,,1■2}2■2} has recursion level C(C(Ω22+C(C(Ω22,0),a)+C(C(Ω2+C(Ω2+1,a),a),a),0),0)
  • {1,,1{1{1,,1■3}2,,1■2}1{1{1,,1■3}2,,1■2}2■2} has recursion level C(C(Ω22+C(C(Ω22,0),a)2,0),0)
  • {1,,1{2{1,,1■3}2,,1■2}2■2} has recursion level C(C(Ω22+ω^(C(C(Ω22,0),a)+1),0),0)
  • {1,,1{1{1,,2■2}2{1,,1■3}2,,1■2}2■2} has recursion level C(C(Ω22+C(C(Ω2,a),C(C(Ω22,0),a)),0),0)
  • {1,,1{1{1,,3■2}2{1,,1■3}2,,1■2}2■2} has recursion level C(C(Ω22+C(C(Ω2,C(Ω2,a)),C(C(Ω22,0),a)),0),0)
  • {1,,1{1{1,,1{1,,1■2}2■2}2{1,,1■3}2,,1■2}2■2} has recursion level C(C(Ω22+C(C(Ω2+a,a),C(C(Ω22,0),a)),0),0)
  • {1,,1{1{1,,1{1,,2■2}2■2}2{1,,1■3}2,,1■2}2■2} has recursion level C(C(Ω22+C(C(Ω2+C(Ω2,a),a),C(C(Ω22,0),a)),0),0)
  • {1,,1{1{1,,1{1,,1{1,,1■2}2■2}2■2}2{1,,1■3}2,,1■2}2■2} has recursion level C(C(Ω22+C(C(Ω2+C(Ω2+a,a),a),C(C(Ω22,0),a)),0),0)
  • {1,,1{1{1,,1■3}3,,1■2}2■2} has recursion level C(C(Ω22+C(C(Ω22,0),C(C(Ω22,0),a)),0),0)
  • {1,,1{1{1,,1■3}1,2,,1■2}2■2} has recursion level C(C(Ω22+C(C(Ω22,0)+1,a),0),0)
  • {1,,1{1{1,,1■3}1{1,,2■2}2,,1■2}2■2} has recursion level C(C(Ω22+C(C(Ω22,0)+C(Ω2,a),a),0),0)
  • {1,,1{1{1,,1■3}1{1,,1{1,,1■2}2■2}2,,1■2}2■2} has recursion level C(C(Ω22+C(C(Ω22,0)+C(C(Ω2+a,a),C(Ω2,a)),a),0),0)
  • {1,,1{1{1,,1■3}1{1,,1{1,,1{1,,1■2}2■2}2■2}2,,1■2}2■2} has recursion level C(C(Ω22+C(C(Ω22,0)+C(C(Ω2+C(Ω2+a,a),a),C(Ω2,a)),a),0),0)
  • {1,,1{1{1,,1■3}1{1,,1■3}2,,1■2}2■2} has recursion level C(C(Ω22+C(C(Ω22,0)+C(C(Ω22,0),C(Ω2,a)),a),0),0)
  • {1,,1{1{2,,1■3}2,,1■2}2■2} has recursion level C(C(Ω22+C(C(Ω22,0)+ω^(C(C(Ω22,0),C(Ω2,a))+1),a),0),0)
  • {1,,1{1{1{1,,2■2}2,,1■3}2,,1■2}2■2} has recursion level C(C(Ω22+C(C(Ω22,0)+ω^(C(C(Ω22,0),C(Ω2,a))+C(Ω2,a)),a),0),0)
  • {1,,1{1{1{1,,1■3}2,,2■2}2,,1■2}2■2} has recursion level C(C(Ω22+C(C(Ω22,0),a),0),0)
  • {1,,1{1{1{1,,1■3}2,,2■2}1,2,,1■2}2■2} has recursion level C(C(Ω22+C(C(Ω22,0)+1,a),0),0)
  • {1,,1{1{1{1,,1■3}2,,2■2}1{1{1,,1■3}2,,2■2}2,,1■2}2■2} has recursion level C(C(Ω22+C(C(Ω22,0)+C(C(Ω22,0),C(Ω2,a)),a),0),0)
  • {1,,1{1{2{1,,1■3}2,,2■2}2,,1■2}2■2} has recursion level C(C(Ω22+C(C(Ω22,0)+ω^(C(C(Ω22,0),C(Ω2,a))+1),a),0),0)
  • {1,,1{1{1{1,,3■2}2{1,,1■3}2,,2■2}2,,1■2}2■2} has recursion level C(C(Ω22+C(C(Ω22,0)+C(C(Ω2,C(Ω2,a)),C(C(Ω22,0),C(Ω2,a))),a),0),0)
  • {1,,1{1{1{1,,1{1,,1■2}2■2}2{1,,1■3}2,,2■2}2,,1■2}2■2} has recursion level C(C(Ω22+C(C(Ω22,0)+C(C(Ω2+a,a),C(C(Ω22,0),C(Ω2,a))),a),0),0)
  • {1,,1{1{1{1,,1■3}3,,2■2}2,,1■2}2■2} has recursion level C(C(Ω22+C(C(Ω22,0)+C(C(Ω22,0),C(C(Ω22,0),C(Ω2,a))),a),0),0)
  • {1,,1{1{1{1,,1■3}1,2,,2■2}2,,1■2}2■2} has recursion level C(C(Ω22+C(C(Ω22,0)+C(C(Ω22,0)+1,C(Ω2,a)),a),0),0)
  • {1,,1{1{1{1,,1■3}1{1,,3■2}2,,2■2}2,,1■2}2■2} has recursion level C(C(Ω22+C(C(Ω22,0)+C(Ω2,C(Ω2,a)),a),0),0)
  • {1,,1{1{1{1,,1■3}1{1,,3■2}1,2,,2■2}2,,1■2}2■2} has recursion level C(C(Ω22+C(C(Ω22,0)+C(Ω2,C(Ω2,a))+1,a),0),0)
  • {1,,1{1{1{1,,1■3}1{2,,3■2}2,,2■2}2,,1■2}2■2} has recursion level C(C(Ω22+C(C(Ω22,0)+ω^(C(Ω2,C(Ω2,a))+1),a),0),0)
  • {1,,1{1{1{1,,1■3}1{1,,1■3}2,,2■2}2,,1■2}2■2} has recursion level C(C(Ω22+C(C(Ω22,0)+C(C(Ω22,0),C(Ω2,C(Ω2,a))),a),0),0)
  • {1,,1{1{1{2,,1■3}2,,2■2}2,,1■2}2■2} has recursion level C(C(Ω22+C(C(Ω22,0)+ω^(C(C(Ω22,0),C(Ω2,C(Ω2,a)))+1),a),0),0)
  • {1,,1{1{1{1{1,,3■2}2,,1■3}2,,2■2}2,,1■2}2■2} has recursion level C(C(Ω22+C(C(Ω22,0)+ω^(C(C(Ω22,0),C(Ω2,C(Ω2,a)))+C(Ω2,C(Ω2,a))),a),0),0)

So {1,,1{1{1{1,,3■2}2,,1■3}2,,1■2}2■2} has recursion level C(C(Ω22+C(C(Ω22,0)+ω^(C(C(Ω22,0),C(Ω2,C(Ω2,a)))+C(Ω2,C(Ω2,a))),a),0),0). And further, the recursion level of {1,,1{1{1{1,,k■2}2,,1■3}2,,1■2}2■2} is between C(C(Ω22+C(C(Ω22,0)+C(Ω2,C(Ω2,…C(Ω2,a))),a),0),0) (with k-1 Ω2′s in the blue part) and C(C(Ω22+C(C(Ω22,0)+C(Ω2,C(Ω2,…C(Ω2,a))),a),0),0) (with k Ω2′s in the blue part), so {1,,1{1{1{1,,1,2■2}2,,1■3}2,,1■2}2■2} has recursion level C(C(Ω22+C(C(Ω22,0)+C(Ω2+1,a),a),0),0).

  • {1,,1{1{1{1,,1{1,,1■2}2■2}2,,1■3}2,,1■2}2■2} has recursion level C(C(Ω22+C(C(Ω22,0)+C(Ω2+a,a),a),0),0)
  • {1{1,,1{1,,2■2}2■2}2,,1■3} has recursion level C(C(Ω22+C(C(Ω22,0)+C(Ω2+C(Ω2,a),a),a),0),0)
  • {1{1,,1{1,,1{1,,1■2}2■2}2■2}2,,1■3} has recursion level C(C(Ω22+C(C(Ω22,0)+C(Ω2+C(Ω2+a,a),a),a),0),0)
  • {1{1,,1■3}2,,1■3} has recursion level C(C(Ω22+C(C(Ω22,0)2,a),0),0)
  • {1,,1{1{1{1,,1,2■2}2{1,,1■3}2,,1■3}2,,1■2}2■2} has recursion level C(C(Ω22+C(C(Ω22,0)2+C(Ω2+1,a),a),0),0)
  • {1{1,,1■3}3,,1■3} has recursion level C(C(Ω22+C(C(Ω22,0)3,a),0),0)
  • {1{1,,1■3}1,2,,1■3} has recursion level C(C(Ω22+C(ω^(C(Ω22,0)+1),a),0),0)
  • {1{1,,1■3}1{1,,1■3}2,,1■3} has recursion level C(C(Ω22+C(ω^(C(Ω22,0)2),a),0),0)
  • {1{2,,1■3}2,,1■3} has recursion level C(C(Ω22+C(ω^ω^(C(Ω22,0)+1),a),0),0)
  • {1{1,,2■3}2,,1■3} has recursion level C(C(Ω22+C(C(Ω2,C(Ω22,0)),a),0),0)
  • also, {1,,2■3} has recursion level C(C(Ω22+C(C(Ω2,C(Ω22,0)),a),0),0)
  • {1,,1{1{1{1,,1,2■2}2{1,,2■3}2,,1■3}2,,1■2}2■2} has recursion level C(C(Ω22+C(C(Ω2,C(Ω22,0))+C(Ω2+1,a),a),0),0)
  • {1{1,,1■3}2{1,,2■3}2,,1■3} has recursion level C(C(Ω22+C(C(Ω2,C(Ω22,0))+C(Ω22,0),a),0),0)
  • {1{1{1,,2■3}2,,1■3}2{1,,2■3}2,,1■3} has recursion level C(C(Ω22+C(C(Ω2,C(Ω22,0))+C(C(Ω2,C(Ω22,0)),C(Ω22,0)),a),0),0)
  • {1{1,,2■3}3,,1■3} has recursion level C(C(Ω22+C(C(Ω2,C(Ω22,0))2,a),0),0)
  • {1{1,,2■3}1,2,,1■3} has recursion level C(C(Ω22+C(ω^(C(Ω2,C(Ω22,0))+1),a),0),0)
  • {1{2,,2■3}2,,1■3} has recursion level C(C(Ω22+C(ω^ω^(C(Ω2,C(Ω22,0))+1),a),0),0)
  • {1,,3■3} has recursion level C(C(Ω22+C(C(Ω2,C(Ω2,C(Ω22,0))),a),0),0)
  • {1,,1,2■3} has recursion level C(C(Ω22+C(C(Ω2+1,C(Ω22,0)),a),0),0)
  • {1,,1{1,,1■2}2■3} has recursion level C(C(Ω22+C(C(Ω2+a,C(Ω22,0)),a),0),0)
  • {1,,1{1,,2■2}2■3} has recursion level C(C(Ω22+C(C(Ω2+C(Ω2,a),C(Ω22,0)),a),0),0)
  • {1,,1{1,,1{1,,1■2}2■2}2■3} has recursion level C(C(Ω22+C(C(Ω2+C(Ω2+a,a),C(Ω22,0)),a),0),0)
  • {1,,1{1,,1■3}2■3} has recursion level C(C(Ω22+C(C(Ω22,C(Ω22,0)),a),0),0) (here again, C(C(Ω2+C(Ω22,0),C(Ω22,0)),a) is not a valid term)
  • {1,,2{1,,1■3}2■3} has recursion level C(C(Ω22+C(C(Ω2,C(Ω22,C(Ω22,0))),a),0),0)
  • {1,,1{1,,1■3}3■3} has recursion level C(C(Ω22+C(C(Ω22,C(Ω22,C(Ω22,0))),a),0),0)
  • {1,,1{1,,1■3}1,2■3} has recursion level C(C(Ω22+C(C(Ω22+1,0),a),0),0)
  • {1,,1{1,,1■3}1{1,,1■2}2■3} has recursion level C(C(Ω22+C(C(Ω22+a,0),a),0),0)
  • {1,,1{1,,1■3}1{1{1,,1{1,,1■2}2■2}2,,1■2}2■3} has recursion level C(C(Ω22+C(C(Ω22+C(C(Ω2+a,a),a),0),a),0),0)
  • {1,,1{1,,1■3}1{1{1,,1■3}2,,1■2}2■3} has recursion level C(C(Ω22+C(C(Ω22+C(C(Ω22,0),a),0),a),0),0)
  • {1,,1{1,,1■3}1{1{1,,1{1,,1■3}1,2■3}2,,1■2}2■3} has recursion level C(C(Ω22+C(C(Ω22+C(C(Ω22+1,0),a),0),a),0),0)
  • {1,,1{1,,1■3}1{1{1,,1{1,,1■3}1{1,,1■2}2■3}2,,1■2}2■3} has recursion level C(C(Ω22+C(C(Ω22+C(C(Ω22+a,0),a),0),a),0),0)
  • {1,,1{1,,1■3}1{1,,2■2}2■3} has recursion level C(C(Ω22+C(Ω2,a),0),0) (here, it hits the line of “C(C(Ω22+α,0),0) is valid”)
  • also, {1,,1{1{1,,1{1,,1■3}1{1,,2■2}2■3}2,,1■2}2■2} has recursion level C(C(Ω22+C(Ω2,a),0),0)
  • {1,,1{1{1,,1{1,,1■3}1{1,,2■2}2■3}2,,1■2}1,2■2} has recursion level C(C(Ω22+C(Ω2,a)+1,0),0)
  • {1,,1{1{1,,1{1,,1■3}1{1,,2■2}2■3}2,,1■2}1{1,,1■2}2■2} has recursion level C(C(Ω22+C(Ω2,a)+a,0),0)
  • {1,,1{1{1,,1{1,,1■3}1{1,,2■2}2■3}2,,1■2}1{1{1,,1■3}2,,1■2}2■2} has recursion level C(C(Ω22+C(Ω2,a)+C(C(Ω22,0),a),0),0)
  • {1,,1{1{1,,1{1,,1■3}1{1,,2■2}2■3}2,,1■2}1{1{1,,1{1,,1■3}1{1,,1■2}2■3}2,,1■2}2■2} has recursion level C(C(Ω22+C(Ω2,a)+C(C(Ω22+a,0),a),0),0)
  • {1,,1{1{1,,1{1,,1■3}1{1,,2■2}2■3}2,,1■2}1{1{1,,1{1,,1■3}1{1,,2■2}2■3}2,,1■2}2■2} has recursion level C(C(Ω22+C(Ω2,a)+C(C(Ω22+C(Ω2,a),0),a),0),0)
  • {1,,1{2{1,,1{1,,1■3}1{1,,2■2}2■3}2,,1■2}2■2} has recursion level C(C(Ω22+C(Ω2,a)+ω^(C(C(Ω22+C(Ω2,a),0),a)+1),0),0)
  • {1,,1{1{1,,1■3}2{1,,1{1,,1■3}1{1,,2■2}2■3}2,,1■2}2■2} has recursion level C(C(Ω22+C(Ω2,a)+C(C(Ω22,0),C(C(Ω22+C(Ω2,a),0),a)),0),0)
  • {1,,1{1{1,,1{1,,1■3}1{1,,1■2}2■3}2{1,,1{1,,1■3}1{1,,2■2}2■3}2,,1■2}2■2} has recursion level C(C(Ω22+C(Ω2,a)+C(C(Ω22+a,0),C(C(Ω22+C(Ω2,a),0),a)),0),0)
  • {1,,1{1{1,,1{1,,1■3}1{1,,2■2}2■3}3,,1■2}2■2} has recursion level C(C(Ω22+C(Ω2,a)+C(C(Ω22+C(Ω2,a),0),C(C(Ω22+C(Ω2,a),0),a)),0),0)
  • {1,,1{1{1,,1{1,,1■3}1{1,,2■2}2■3}1,2,,1■2}2■2} has recursion level C(C(Ω22+C(Ω2,a)+C(C(Ω22+C(Ω2,a),0)+1,a),0),0)
  • {1,,1{1{1,,1{1,,1■3}1{1,,2■2}2■3}1{1,,2■2}2,,1■2}2■2} has recursion level C(C(Ω22+C(Ω2,a)+C(C(Ω22+C(Ω2,a),0)+C(Ω2,a),a),0),0)
  • {1,,1{1{1{1,,1,2■2}2,,1{1,,1■3}1{1,,2■2}2■3}2,,1■2}2■2} has recursion level C(C(Ω22+C(Ω2,a)+C(C(Ω22+C(Ω2,a),0)+C(Ω2+1,a),a),0),0)
  • {1{1,,1■3}2,,1{1,,1■3}1{1,,2■2}2■3} has recursion level C(C(Ω22+C(Ω2,a)+C(C(Ω22+C(Ω2,a),0)+C(Ω22,0),a),0),0)
  • {1{1,,1{1,,1■3}1{1,,1■2}2■3}2,,1{1,,1■3}1{1,,2■2}2■3} has recursion level C(C(Ω22+C(Ω2,a)+C(C(Ω22+C(Ω2,a),0)+C(Ω22+a,0),a),0),0)
  • {1{1,,1{1,,1■3}1{1,,2■2}2■3}2,,1{1,,1■3}1{1,,2■2}2■3} has recursion level C(C(Ω22+C(Ω2,a)+C(C(Ω22+C(Ω2,a),0)2,a),0),0)
  • {1{1,,1{1,,1■3}1{1,,2■2}2■3}1,2,,1{1,,1■3}1{1,,2■2}2■3} has recursion level C(C(Ω22+C(Ω2,a)+C(ω^(C(Ω22+C(Ω2,a),0)+1),a),0),0)
  • {1{2,,1{1,,1■3}1{1,,2■2}2■3}2,,1{1,,1■3}1{1,,2■2}2■3} has recursion level C(C(Ω22+C(Ω2,a)+C(ω^ω^(C(Ω22+C(Ω2,a),0)+1),a),0),0)
  • {1,,2{1,,1■3}1{1,,2■2}2■3} has recursion level C(C(Ω22+C(Ω2,a)+C(C(Ω2,C(Ω22+C(Ω2,a),0)),a),0),0)
  • {1,,1{1,,1■2}2{1,,1■3}1{1,,2■2}2■3} has recursion level C(C(Ω22+C(Ω2,a)+C(C(Ω2+a,C(Ω22+C(Ω2,a),0)),a),0),0)
  • {1,,1{1,,1{1,,1■2}2■2}2{1,,1■3}1{1,,2■2}2■3} has recursion level C(C(Ω22+C(Ω2,a)+C(C(Ω2+C(Ω2+a,a),C(Ω22+C(Ω2,a),0)),a),0),0)
  • {1,,1{1,,1■3}2{1,,2■2}2■3} has recursion level C(C(Ω22+C(Ω2,a)+C(C(Ω22,C(Ω22+C(Ω2,a),0)),a),0),0)
  • {1,,1{1,,1■3}1{1,,1■2}2{1,,2■2}2■3} has recursion level C(C(Ω22+C(Ω2,a)+C(C(Ω22+a,C(Ω22+C(Ω2,a),0)),a),0),0)
  • {1,,1{1,,1■3}1{1,,2■2}3■3} has recursion level C(C(Ω22+C(Ω2,a)+C(C(Ω22+C(Ω2,a),C(Ω22+C(Ω2,a),0)),a),0),0)
  • {1,,1{1,,1■3}1{1,,2■2}1,2■3} has recursion level C(C(Ω22+C(Ω2,a)+C(C(Ω22+C(Ω2,a)+1,0),a),0),0)
  • {1,,1{1,,1■3}1{1,,2■2}1{1,,1■2}2■3} has recursion level C(C(Ω22+C(Ω2,a)+C(C(Ω22+C(Ω2,a)+a,0),a),0),0)
  • {1,,1{1,,1■3}1{1,,2■2}1{1{1,,1{1,,1■3}1{1,,2■2}1{1,,1■2}2■3}2,,1■2}2■3} has recursion level C(C(Ω22+C(Ω2,a)+C(C(Ω22+C(Ω2,a)+C(C(Ω22+C(Ω2,a)+a,0),a),0),a),0),0)
  • {1,,1{1,,1■3}1{1,,2■2}1{1,,2■2}2■3} has recursion level C(C(Ω22+C(Ω2,a)2,0),0)
  • {1,,1{1,,1■3}1{2,,2■2}2■3} has recursion level C(C(Ω22+ω^(C(Ω2,a)+1),0),0)
  • {1,,1{1,,1■3}1{1{1,,3■2}2,,2■2}2■3} has recursion level C(C(Ω22+C(C(Ω2,C(Ω2,a)),C(Ω2,a)),0),0)
  • {1,,1{1,,1■3}1{1{1,,1,2■2}2,,2■2}2■3} has recursion level C(C(Ω22+C(C(Ω2+1,a),C(Ω2,a)),0),0)
  • {1,,1{1,,1■3}1{1{1,,1{1,,1,2■2}2■2}2,,2■2}2■3} has recursion level C(C(Ω22+C(C(Ω2+C(Ω2+1,a),a),C(Ω2,a)),0),0)
  • {1,,1{1,,1■3}1{1{1,,1■3}2,,2■2}2■3} has recursion level C(C(Ω22+C(C(Ω22,0),C(Ω2,a)),0),0)
  • {1,,1{1,,1■3}1{1{1,,1{1,,1■3}1{1,,1■2}2■3}2,,2■2}2■3} has recursion level C(C(Ω22+C(C(Ω22+a,0),C(Ω2,a)),0),0)
  • {1,,1{1,,1■3}1{1{1,,1{1,,1■3}1{1,,2■2}2■3}2,,2■2}2■3} has recursion level C(C(Ω22+C(C(Ω22+C(Ω2,a),0),C(Ω2,a)),0),0)
  • {1,,1{1,,1■3}1{1{1,,1{1,,1■3}1{1{1,,1{1,,1■3}1{1,,2■2}2■3}2,,2■2}2■3}2,,2■2}2■3} has recursion level C(C(Ω22+C(C(Ω22+C(C(Ω22+C(Ω2,a),0),C(Ω2,a)),0),C(Ω2,a)),0),0)
  • {1,,1{1,,1■3}1{1,,3■2}2■3} has recursion level C(C(Ω22+C(Ω2,C(Ω2,a)),0),0)
  • {1,,1{1,,1■3}1{1,,4■2}2■3} has recursion level C(C(Ω22+C(Ω2,C(Ω2,C(Ω2,a))),0),0)
  • {1,,1{1,,1■3}1{1,,1,2■2}2■3} has recursion level C(C(Ω22+C(Ω2+1,a),0),0)
  • {1,,1{1,,1■3}1{1,,1{1,,1■2}2■2}2■3} has recursion level C(C(Ω22+C(Ω2+a,a),0),0)
  • {1,,1{1,,1■3}1{1,,1{1,,1{1,,1■2}2■2}2■2}2■3} has recursion level C(C(Ω22+C(Ω2+C(Ω2+a,a),a),0),0)
  • {1,,1{1,,1■3}1{1,,1■3}2■3} has recursion level C(C(Ω22+C(Ω2+C(Ω22,0),a),0),0)

It seems that {1,,1■3} approximately corresponds to C(Ω22,0), but it’s not so in following comparisons. Now let ordinal a = C(Ω2+C(Ω22,0),0) and b = C(Ω2+C(Ω22,0),a) = C(Ω2+C(Ω22,0),C(Ω2+C(Ω22,0),0)).

  • {1,,1{1,,1■3}1{1,,1■3}2■3} has recursion level C(C(Ω22+b,0),0)
  • {1,,1{1{1,,1{1,,1■3}1{1,,1■3}2■3}2,,1■2}1,2■2} has recursion level C(C(Ω22+b+1,0),0)
  • {1,,1{1{1,,1{1,,1■3}1{1,,1■3}2■3}2,,1■2}1{1,,1■2}2■2} has recursion level C(C(Ω22+b+a,0),0)
  • {1,,1{1{1,,1{1,,1■3}1{1,,1■3}2■3}2,,1■2}1{1{1,,1{1,,1■3}1{1,,1■3}2■3}2,,1■2}2■2} has recursion level C(C(Ω22+b+C(C(Ω22+b,0),a),0),0)
  • {1,,1{2{1,,1{1,,1■3}1{1,,1■3}2■3}2,,1■2}2■2} has recursion level C(C(Ω22+b+ω^(C(C(Ω22+b,0),a)+1),0),0)
  • {1,,1{1{1,,1{1,,1■3}1{1,,1■3}2■3}1,2,,1■2}2■2} has recursion level C(C(Ω22+b+C(C(Ω22+b,0)+1,a),0),0)
  • {1{1,,1■3}2,,1{1,,1■3}1{1,,1■3}2■3} has recursion level C(C(Ω22+b+C(C(Ω22+b,0)+C(Ω22,0),a),0),0)
  • {1{1,,1{1,,1■3}1{1,,1■3}2■3}2,,1{1,,1■3}1{1,,1■3}2■3} has recursion level C(C(Ω22+b+C(C(Ω22+b,0)2,a),0),0)
  • {1,,2{1,,1■3}1{1,,1■3}2■3} has recursion level C(C(Ω22+b+C(C(Ω2,C(Ω22+b,0)),a),0),0)
  • {1,,1{1,,1■3}2{1,,1■3}2■3} has recursion level C(C(Ω22+b+C(C(Ω22,C(Ω22+b,0)),a),0),0)
  • {1,,1{1,,1■3}1{1,,1■3}1,2■3} has recursion level C(C(Ω22+b+C(C(Ω22+b+1,0),a),0),0)
  • {1,,1{1,,1■3}1{1,,1■3}1{1,,1■2}2■3} has recursion level C(C(Ω22+b+C(C(Ω22+b+a,0),a),0),0)
  • {1,,1{1,,1■3}1{1,,1■3}1{1,,2■2}2■3} has recursion level C(C(Ω22+b+C(Ω2,a),0),0)
  • {1,,1{1,,1■3}1{1,,1■3}1{1,,1{1,,2■2}2■2}2■3} has recursion level C(C(Ω22+b+C(Ω2+C(Ω2,a),a),0),0)
  • {1,,1{1,,1■3}1{1,,1■3}1{1,,1■3}2■3} has recursion level C(C(Ω22+b2,0),0)
  • {1,,1{2,,1■3}2■3} has recursion level C(C(Ω22+ω^(b+1),0),0)
  • {1,,1{1{1,,1■2}2,,1■3}2■3} has recursion level C(C(Ω22+ω^(b+a),0),0)
  • {1,,1{1{1,,2■2}2,,1■3}2■3} has recursion level C(C(Ω22+ω^(b+C(Ω2,a)),0),0)
  • {1,,1{1{1,,1{1,,2■2}2■2}2,,1■3}2■3} has recursion level C(C(Ω22+ω^(b+C(Ω2+C(Ω2,a),a)),0),0)
  • {1,,1{1{1,,1■3}2,,1■3}2■3} has recursion level C(C(Ω22+ω^(b2),0),0)
  • {1,,1{1{1,,2■3}2,,1■3}2■3} has recursion level C(C(Ω22+C(C(Ω2,b),b),0),0)
  • {1,,1{1{1,,1,2■3}2,,1■3}2■3} has recursion level C(C(Ω22+C(C(Ω2+1,b),b),0),0)
  • {1,,1{1{1,,1{1,,1■3}2■3}2,,1■3}2■3} has recursion level C(C(Ω22+C(C(Ω2+b,b),b),0),0)
  • {1,,1{1{1,,1{2,,1■3}2■3}2,,1■3}2■3} has recursion level C(C(Ω22+C(C(Ω2+ω^(b+1),b),b),0),0)
  • {1,,1{1{1,,1{1{1,,1{1,,1■3}2■3}2,,1■3}2■3}2,,1■3}2■3} has recursion level C(C(Ω22+C(C(Ω2+C(C(Ω2+b,b),b),b),b),0),0)
  • {1,,1{1,,2■3}2■3} has recursion level C(C(Ω22+C(C(Ω2+C(Ω2,b),b),b),0),0)
  • {1,,1{1,,3■3}2■3} has recursion level C(C(Ω22+C(C(Ω2+C(Ω2,C(Ω2,b)),b),b),0),0)
  • {1,,1{1,,1,2■3}2■3} has recursion level C(C(Ω22+C(C(Ω2+C(Ω2+1,b),b),b),0),0)
  • {1,,1{1,,1{1,,1■3}2■3}2■3} has recursion level C(C(Ω22+C(C(Ω2+C(Ω2+b,b),b),b),0),0)
  • {1,,1■4} has recursion level C(C(Ω22+C(C(Ω22,0),b),0),0)
  • {1,,1{1{1,,1■4}2,,1■3}1{1,,1■3}2■3} has recursion level C(C(Ω22+C(C(Ω22,0),b)+b,0),0)
  • {1,,1{1{1,,1■4}2,,1■3}1{1{1,,2■3}2,,1■3}2■3} has recursion level C(C(Ω22+C(C(Ω22,0),b)+C(C(Ω2,b),b),0),0)
  • {1,,1{1{1,,1■4}2,,1■3}1{1{1,,1{1,,1■3}2■3}2,,1■3}2■3} has recursion level C(C(Ω22+C(C(Ω22,0),b)+C(C(Ω2+b,b),b),0),0)
  • {1,,1{1{1,,1■4}2,,1■3}1{1{1,,1■4}2,,1■3}2■3} has recursion level C(C(Ω22+C(C(Ω22,0),b)2,0),0)
  • {1,,1{2{1,,1■4}2,,1■3}2■3} has recursion level C(C(Ω22+ω^(C(C(Ω22,0),b)+1),0),0)
  • {1,,1{1{1,,2■3}2{1,,1■4}2,,1■3}2■3} has recursion level C(C(Ω22+C(C(Ω2,b),C(C(Ω22,0),b)),0),0)
  • {1,,1{1{1,,1{1,,1■3}2■3}2{1,,1■4}2,,1■3}2■3} has recursion level C(C(Ω22+C(C(Ω2+b,b),C(C(Ω22,0),b)),0),0)
  • {1,,1{1{1,,1■4}3,,1■3}2■3} has recursion level C(C(Ω22+C(C(Ω22,0),C(C(Ω22,0),b)),0),0)
  • {1,,1{1{1,,1■4}1,2,,1■3}2■3} has recursion level C(C(Ω22+C(C(Ω22,0)+1,b),0),0)
  • {1,,1{1{1,,1■4}1{1,,2■3}2,,1■3}2■3} has recursion level C(C(Ω22+C(C(Ω22,0)+C(Ω2,b),b),0),0)
  • {1,,1{1{1,,1■4}1{1,,1{1,,1■3}2■3}2,,1■3}2■3} has recursion level C(C(Ω22+C(C(Ω22,0)+C(C(Ω2+b,b),C(Ω2,b)),b),0),0)
  • {1,,1{1{1,,1■4}1{1,,1■4}2,,1■3}2■3} has recursion level C(C(Ω22+C(C(Ω22,0)+C(C(Ω22,0),C(Ω2,b)),b),0),0)
  • {1,,1{1{2,,1■4}2,,1■3}2■3} has recursion level C(C(Ω22+C(C(Ω22,0)+ω^(C(C(Ω22,0),C(Ω2,b))+1),b),0),0)
  • {1,,1{1{1{1,,2■3}2,,1■4}2,,1■3}2■3} has recursion level C(C(Ω22+C(C(Ω22,0)+ω^(C(C(Ω22,0),C(Ω2,b))+C(Ω2,b)),b),0),0)
  • {1,,1{1{1{1,,3■3}2,,1■4}2,,1■3}2■3} has recursion level C(C(Ω22+C(C(Ω22,0)+ω^(C(C(Ω22,0),C(Ω2,C(Ω2,b)))+C(Ω2,C(Ω2,b))),b),0),0)
  • {1,,1{1{1{1,,1,2■3}2,,1■4}2,,1■3}2■3} has recursion level C(C(Ω22+C(C(Ω22,0)+C(Ω2+1,b),b),0),0)
  • {1{1,,1■4}2,,1■4} has recursion level C(C(Ω22+C(C(Ω22,0)2,b),0),0)
  • {1{1,,1■4}1,2,,1■4} has recursion level C(C(Ω22+C(ω^(C(Ω22,0)+1),b),0),0)
  • {1{1,,1■4}1{1,,1■4}2,,1■4} has recursion level C(C(Ω22+C(ω^(C(Ω22,0)2),b),0),0)
  • {1{2,,1■4}2,,1■4} has recursion level C(C(Ω22+C(ω^ω^(C(Ω22,0)+1),b),0),0)
  • {1,,2■4} has recursion level C(C(Ω22+C(C(Ω2,C(Ω22,0)),b),0),0)
  • {1,,1,2■4} has recursion level C(C(Ω22+C(C(Ω2+1,C(Ω22,0)),b),0),0)
  • {1,,1{1,,1■2}2■4} has recursion level C(C(Ω22+C(C(Ω2+a,C(Ω22,0)),b),0),0)
  • {1,,1{1,,2■2}2■4} has recursion level C(C(Ω22+C(C(Ω2+C(Ω2,a),C(Ω22,0)),b),0),0)
  • {1,,1{1,,1■3}2■4} has recursion level C(C(Ω22+C(C(Ω2+b,C(Ω22,0)),b),0),0)
  • {1,,1{1,,2■3}2■4} has recursion level C(C(Ω22+C(C(Ω2+C(Ω2,b),C(Ω22,0)),b),0),0)
  • {1,,1{1,,1■4}2■4} has recursion level C(C(Ω22+C(C(Ω22,C(Ω22,0)),b),0),0)
  • {1,,1{1,,1■4}1,2■4} has recursion level C(C(Ω22+C(C(Ω22+1,0),b),0),0)
  • {1,,1{1,,1■4}1{1,,1■3}2■4} has recursion level C(C(Ω22+C(C(Ω22+b,0),b),0),0)
  • {1,,1{1,,1■4}1{1,,2■3}2■4} has recursion level C(C(Ω22+C(Ω2,b),0),0)
  • {1,,1{1,,1■4}1{1,,3■3}2■4} has recursion level C(C(Ω22+C(Ω2,C(Ω2,b)),0),0)
  • {1,,1{1,,1■4}1{1,,1,2■3}2■4} has recursion level C(C(Ω22+C(Ω2+1,b),0),0)
  • {1,,1{1,,1■4}1{1,,1{1,,1■3}2■3}2■4} has recursion level C(C(Ω22+C(Ω2+b,b),0),0)
  • {1,,1{1,,1■4}1{1,,1{1,,1{1,,1■3}2■3}2■3}2■4} has recursion level C(C(Ω22+C(Ω2+C(Ω2+b,b),b),0),0)
  • {1,,1{1,,1■4}1{1,,1■4}2■4} has recursion level C(C(Ω22+C(Ω2+C(Ω22,0),b),0),0)
  • {1,,1{1{1,,2■4}2,,1■4}2■4} has recursion level C(C(Ω22+C(C(Ω2,C(Ω2+C(Ω22,0),b)),C(Ω2+C(Ω22,0),b)),0),0)
  • {1,,1■5} has recursion level C(C(Ω22+C(C(Ω22,0),C(Ω2+C(Ω22,0),b)),0),0)
  • {1,,1{1,,1■5}1{1,,2■4}2■5} has recursion level C(C(Ω22+C(Ω2,C(Ω2+C(Ω22,0),b)),0),0)
  • {1,,1{1,,1■5}1{1,,1■5}2■5} has recursion level C(C(Ω22+C(Ω2+C(Ω22,0),C(Ω2+C(Ω22,0),b)),0),0)
  • {1,,1■6} has recursion level C(C(Ω22+C(C(Ω22,0),C(Ω2+C(Ω22,0),C(Ω2+C(Ω22,0),b))),0),0)

Generally, the recursion level of {1,,1■k} is between C(C(Ω22+C(Ω2+C(Ω22,0),C(Ω2+C(Ω22,0),…C(Ω2+C(Ω22,0),0))),0),0) (with k-2 Ω2+C(Ω22,0)′s in the blue part) and C(C(Ω22+C(Ω2+C(Ω22,0),C(Ω2+C(Ω22,0),…C(Ω2+C(Ω22,0),0))),0),0) (with k-1 Ω2+C(Ω22,0)′s in the blue part), so {1,,1■1,2} has recursion level C(C(Ω22+C(Ω2+C(Ω22,0)+1,0),0),0). Now let ordinal a = C(Ω2+C(Ω22,0),0) and b = C(Ω2+C(Ω22,0)+1,0).

  • {1,,1{1{1,,1■1,2}2,,1■2}1{1,,1■2}2■2} has recursion level C(C(Ω22+b+a,0),0)
  • {1,,1{1{1,,1■1,2}2,,1■2}1{1{1,,1■3}2,,1■2}2■2} has recursion level C(C(Ω22+b+C(C(Ω22,0),a),0),0)
  • {1,,1{1{1,,1■1,2}2,,1■2}1{1{1,,1■1,2}2,,1■2}2■2} has recursion level C(C(Ω22+b+C(C(Ω22+b,0),a),0),0)
  • {1,,1{1{1,,1■1,2}1,2,,1■2}2■2} has recursion level C(C(Ω22+b+C(C(Ω22+b,0)+1,a),0),0)
  • {1{1,,1■1,2}2,,1■1,2} has recursion level C(C(Ω22+b+C(C(Ω22+b,0)2,a),0),0)
  • {1{1{1,,1■1,2}2,,1■1,2}2,,1■1,2} has recursion level C(C(Ω22+b+C(ω^ω^(C(Ω22+b,0)2),a),0),0)
  • {1,,2■1,2} has recursion level C(C(Ω22+b+C(C(Ω2,C(Ω22+b,0)),a),0),0)
  • {1,,1{1,,1■2}2■1,2} has recursion level C(C(Ω22+b+C(C(Ω2+a,C(Ω22+b,0)),a),0),0)
  • {1,,1{1,,1■3}2■1,2} has recursion level C(C(Ω22+b+C(C(Ω22,C(Ω22+b,0)),a),0),0)
  • {1,,1{1,,1■1,2}2■1,2} has recursion level C(C(Ω22+b+C(C(Ω22+b,C(Ω22+b,0)),a),0),0)
  • {1,,1{1,,1■1,2}1,2■1,2} has recursion level C(C(Ω22+b+C(C(Ω22+b+1,0),a),0),0)
  • {1,,1{1,,1■1,2}1{1,,1■2}2■1,2} has recursion level C(C(Ω22+b+C(C(Ω22+b+a,0),a),0),0)
  • {1,,1{1,,1■1,2}1{1,,2■2}2■1,2} has recursion level C(C(Ω22+b+C(Ω2,a),0),0)
  • {1,,1{1,,1■1,2}1{1,,1■3}2■1,2} has recursion level C(C(Ω22+b+C(Ω2+(Ω22,0),a),0),0)
  • {1,,1{1,,1■1,2}1{1,,1■1,2}2■1,2} has recursion level C(C(Ω22+b2,0),0)
  • {1,,1{1{1,,2■1,2}2,,1■1,2}2■1,2} has recursion level C(C(Ω22+C(C(Ω2,b),b),0),0)
  • {1,,1{1{1,,1{1,,1■1,2}2■1,2}2,,1■1,2}2■1,2} has recursion level C(C(Ω22+C(C(Ω2+b,b),b),0),0)
  • {1,,1{1,,2■1,2}2■1,2} has recursion level C(C(Ω22+C(C(Ω2+C(Ω2,b),b),b),0),0)
  • {1,,1{1,,1{1,,2■1,2}2■1,2}2■1,2} has recursion level C(C(Ω22+C(C(Ω2+C(Ω2+C(Ω2,b),b),b),b),0),0)
  • {1,,1■2,2} has recursion level C(C(Ω22+C(C(Ω22,0),b),0),0)
  • {1{1,,1■2,2}2,,1■2,2} has recursion level C(C(Ω22+C(C(Ω22,0)2,b),0),0)
  • {1,,2■2,2} has recursion level C(C(Ω22+C(C(Ω2,C(Ω22,0)),b),0),0)
  • {1,,1{1,,1■2,2}2■2,2} has recursion level C(C(Ω22+C(C(Ω22,C(Ω22,0)),b),0),0)
  • {1,,1{1,,1■2,2}1{1,,1■1,2}2■2,2} has recursion level C(C(Ω22+C(C(Ω22+b,0),b),0),0)
  • {1,,1{1,,1■2,2}1{1,,2■1,2}2■2,2} has recursion level C(C(Ω22+C(Ω2,b),0),0)
  • {1,,1■3,2} has recursion level C(C(Ω22+C(C(Ω22,0),C(Ω2+C(Ω22,0),b)),0),0)
  • {1,,1■1,3} has recursion level C(C(Ω22+C(Ω2+C(Ω22,0)+1,b),0),0)
  • {1,,1■1,1,2} has recursion level C(C(Ω22+C(Ω2+C(Ω22,0)+2,0),0),0)
  • {1,,1■1{1,,1■2}2} has recursion level C(C(Ω22+C(Ω2+C(Ω22,0)+a,0),0),0)
  • {1,,1■1{1,,1■1,2}2} has recursion level C(C(Ω22+C(Ω2+C(Ω22,0)+b,0),0),0)
  • {1,,1■1■2} has recursion level C(C(Ω22+C(Ω2+C(Ω22,0)2,0),0),0)
  • {1,,1■1,2■2} has recursion level C(C(Ω22+C(Ω2+C(Ω22,0)+1,C(Ω2+C(Ω22,0)2,0)),0),0)
  • {1,,1■1{1,,1■1■2}2■2} has recursion level C(C(Ω22+C(Ω2+C(Ω22,0)+C(Ω2+C(Ω22,0)2,0),C(Ω2+C(Ω22,0)2,0)),0),0)
  • {1,,1■1■3} has recursion level C(C(Ω22+C(Ω2+C(Ω22,0)2,C(Ω2+C(Ω22,0)2,0)),0),0)
  • {1,,1■1■1,2} has recursion level C(C(Ω22+C(Ω2+C(Ω22,0)2+1,0),0),0)
  • {1,,1■1■1■2} has recursion level C(C(Ω22+C(Ω2+C(Ω22,0)3,0),0),0)
  • {1,,1{2,,1,,2}2} has recursion level C(C(Ω22+C(Ω2+ω^(C(Ω22,0)+1),0),0),0)
  • {1,,1{1{1,,1{2,,1,,2}2}2,,1,,2}2} has recursion level C(C(Ω22+C(Ω2+ω^(C(Ω22,0)+C(Ω2+ω^(C(Ω22,0)+1),0)),0),0),0)
  • {1,,1{1■2,,1,,2}2} has recursion level C(C(Ω22+C(Ω2+ω^(C(Ω22,0)2),0),0),0)
  • {1,,1{1■1,2,,1,,2}2} has recursion level C(C(Ω22+C(Ω2+ω^ω^(C(Ω22,0)+1),0),0),0)
  • {1,,1{1{2,,1,,2}2,,1,,2}2} has recursion level C(C(Ω22+C(Ω2+ω^ω^ω^(C(Ω22,0)+1),0),0),0)
  • {1,,1{1{1,,2,,2}2,,1,,2}2} has recursion level C(C(Ω22+C(Ω2+C(C(Ω2,C(Ω22,0)),C(Ω22,0)),0),0),0)

So {1,,2,,2} has recursion level C(C(Ω22+C(Ω2+C(C(Ω2,C(Ω22,0)),C(Ω22,0)),0),0),0). Here’re some approximations, to make the comparisons above more simple:

  • {1,,1■2} (where separator ■ = {1,,1,,2}) approximately corresponds to C(Ω2+C(Ω22,0),0)
  • {1,,2■2} approximately corresponds to C(Ω2,C(Ω2+C(Ω22,0),0))
  • {1,,1■3} approximately corresponds to C(Ω2+C(Ω22,0),C(Ω2+C(Ω22,0),0))
  • {1,,2■3} approximately corresponds to C(Ω2,C(Ω2+C(Ω22,0),C(Ω2+C(Ω22,0),0)))
  • {1,,1■4} approximately corresponds to C(Ω2+C(Ω22,0),C(Ω2+C(Ω22,0),C(Ω2+C(Ω22,0),0)))
  • {1,,1■1,2} approximately corresponds to C(Ω2+C(Ω22,0)+1,0)
  • {1,,1■2,2} approximately corresponds to C(Ω2+C(Ω22,0),C(Ω2+C(Ω22,0)+1,0))
  • {1,,1■1,1,2} approximately corresponds to C(Ω2+C(Ω22,0)+2,0)
  • {1,,1■1■2} approximately corresponds to C(Ω2+C(Ω22,0)2,0)
  • {1,,1{2,,1,,2}2} approximately corresponds to C(Ω2+ω^(C(Ω22,0)+1),0)
  • {1,,1{1■2,,1,,2}2} approximately corresponds to C(Ω2+ω^(C(Ω22,0)2),0)
  • {1,,1{1■1■2,,1,,2}2} approximately corresponds to C(Ω2+ω^ω^(C(Ω22,0)2),0)
  • {1,,1{1{1■2,,1,,2}2,,1,,2}2} approximately corresponds to C(Ω2+ω^ω^ω^(C(Ω22,0)2),0)
  • {1,,1{1,,2,,2}2} approximately corresponds to C(Ω2+C(C(Ω2,C(Ω22,0)),C(Ω22,0)),0)

However, the ordinal C(Ω22,0) in Taranovsky’s ordinal notation behaves in an irregular way.

Up to {1,,1,,3}

  • {1,,1{1,,1,,2}1,2{1{1,,2,,2}2,,1,,2}2} has recursion level C(C(Ω22+C(Ω2+C(Ω22,0)+1,C(Ω2+C(C(Ω2,C(Ω22,0)),C(Ω22,0)),0)),0),0)
  • {1,,1{1,,1,,2}1{1,,1{1{1,,2,,2}2,,1,,2}2}2{1{1,,2,,2}2,,1,,2}2} has recursion level C(C(Ω22+C(Ω2+C(Ω22,0)+C(Ω2+C(C(Ω2,C(Ω22,0)),C(Ω22,0)),0),C(Ω2+C(C(Ω2,C(Ω22,0)),C(Ω22,0)),0)),0),0)
  • {1,,1{1,,1,,2}1{1,,1,,2}2{1{1,,2,,2}2,,1,,2}2} has recursion level C(C(Ω22+C(Ω2+C(Ω22,0)2,C(Ω2+C(C(Ω2,C(Ω22,0)),C(Ω22,0)),0)),0),0)
  • {1,,1{1{1,,1,,2}2,,1,,2}2{1{1,,2,,2}2,,1,,2}2} has recursion level C(C(Ω22+C(Ω2+ω^(C(Ω22,0)2),C(Ω2+C(C(Ω2,C(Ω22,0)),C(Ω22,0)),0)),0),0)
  • {1,,1{1{1,,2,,2}2,,1,,2}3} has recursion level C(C(Ω22+C(Ω2+C(C(Ω2,C(Ω22,0)),C(Ω22,0)),C(Ω2+C(C(Ω2,C(Ω22,0)),C(Ω22,0)),0)),0),0)
  • {1,,1{1{1,,2,,2}2,,1,,2}1,2} has recursion level C(C(Ω22+C(Ω2+C(C(Ω2,C(Ω22,0)),C(Ω22,0))+1,0),0),0)
  • {1,,1{1{1,,2,,2}2,,1,,2}1{1,,1,,2}2} has recursion level C(C(Ω22+C(Ω2+C(C(Ω2,C(Ω22,0)),C(Ω22,0))+C(Ω22,0),0),0),0)
  • {1,,1{1{1,,2,,2}2,,1,,2}1{1{1,,2,,2}2,,1,,2}2} has recursion level C(C(Ω22+C(Ω2+C(C(Ω2,C(Ω22,0)),C(Ω22,0))2,0),0),0)
  • {1{1{1,,2,,2}2,,1,,2}2{1,,2,,2}2,,1,,2} has recursion level C(C(Ω22+C(Ω2+ω^(C(C(Ω2,C(Ω22,0)),C(Ω22,0))2),0),0),0)
  • {1{1,,2,,2}3,,1,,2} has recursion level C(C(Ω22+C(Ω2+C(C(Ω2,C(Ω22,0)),C(C(Ω2,C(Ω22,0)),C(Ω22,0))),0),0),0)
  • {1{1,,2,,2}1,2,,1,,2} has recursion level C(C(Ω22+C(Ω2+C(C(Ω2,C(Ω22,0))+1,C(Ω22,0)),0),0),0)
  • {1{1,,2,,2}1{1,,2,,2}2,,1,,2} has recursion level C(C(Ω22+C(Ω2+C(C(Ω2,C(Ω22,0))2,C(Ω22,0)),0),0),0)
  • {1,,3,,2} has recursion level C(C(Ω22+C(Ω2+C(C(Ω2,C(Ω2,C(Ω22,0))),C(Ω22,0)),0),0),0)
  • {1,,1,2,,2} has recursion level C(C(Ω22+C(Ω2+C(C(Ω2+1,C(Ω22,0)),C(Ω22,0)),0),0),0)
  • {1,,1{1,,1{1,,1,,2}1,2}2,,2} has recursion level C(C(Ω22+C(Ω2+C(C(Ω2+C(Ω2+C(Ω22,0)+1,0),C(Ω22,0)),C(Ω22,0)),0),0),0)
  • {1,,1{1,,1,,2}2,,2} has recursion level C(C(Ω22+C(Ω2+C(C(Ω2+C(Ω22,0),C(Ω22,0)),C(Ω22,0)),0),0),0)
  • {1,,1{1{1,,2,,2}2,,1,,2}2,,2} has recursion level C(C(Ω22+C(Ω2+C(C(Ω2+C(C(Ω2,C(Ω22,0)),C(Ω22,0)),C(Ω22,0)),C(Ω22,0)),0),0),0)
  • {1,,1{1{1,,1{1,,1,,2}2,,2}2,,1,,2}2,,2} has recursion level C(C(Ω22+C(Ω2+C(C(Ω2+C(C(Ω2+C(Ω22,0),C(Ω22,0)),C(Ω22,0)),C(Ω22,0)),C(Ω22,0)),0),0),0)
  • {1,,1{1,,2,,2}2,,2} has recursion level C(C(Ω22+C(Ω2+C(C(Ω22,C(Ω22,0)),C(Ω22,0)),0),0),0) (It’s a bit erratic here. Things like C(Ω2+C(C(Ω2+C(Ω2,C(Ω22,0)),C(Ω22,0)),C(Ω22,0)),0) are not valid)
  • {1,,1{1,,1,,2}1,2{1{1,,1{1,,2,,2}2,,2}2,,1,,2}2} has recursion level C(C(Ω22+C(Ω2+C(Ω22,0)+1,C(Ω2+C(C(Ω22,C(Ω22,0)),C(Ω22,0)),0)),0),0)
  • {1,,1{1{1,,2,,2}2,,1,,2}2{1{1,,1{1,,2,,2}2,,2}2,,1,,2}2} has recursion level C(C(Ω22+C(Ω2+C(C(Ω2,C(Ω22,0)),C(Ω22,0)),C(Ω2+C(C(Ω22,C(Ω22,0)),C(Ω22,0)),0)),0),0)
  • {1,,1{1{1,,1{1,,1,,2}2,,2}2,,1,,2}2{1{1,,1{1,,2,,2}2,,2}2,,1,,2}2} has recursion level C(C(Ω22+C(Ω2+C(C(Ω2+C(Ω22,0),C(Ω22,0)),C(Ω22,0)),C(Ω2+C(C(Ω22,C(Ω22,0)),C(Ω22,0)),0)),0),0)
  • {1,,1{1{1,,1{1,,2,,2}2,,2}2,,1,,2}3} has recursion level C(C(Ω22+C(Ω2+C(C(Ω22,C(Ω22,0)),C(Ω22,0)),C(Ω2+C(C(Ω22,C(Ω22,0)),C(Ω22,0)),0)),0),0)
  • {1,,1{1{1,,1{1,,2,,2}2,,2}2,,1,,2}1,2} has recursion level C(C(Ω22+C(Ω2+C(C(Ω22,C(Ω22,0)),C(Ω22,0))+1,0),0),0)
  • {1,,1{1{1,,1{1,,2,,2}2,,2}2,,1,,2}1{1{1,,1{1,,2,,2}2,,2}2,,1,,2}2} has recursion level C(C(Ω22+C(Ω2+C(C(Ω22,C(Ω22,0)),C(Ω22,0))2,0),0),0)
  • {1{1,,2,,2}2{1,,1{1,,2,,2}2,,2}2,,1,,2} has recursion level C(C(Ω22+C(Ω2+C(C(Ω2,C(Ω22,0)),C(C(Ω22,C(Ω22,0)),C(Ω22,0))),0),0),0)
  • {1{1,,1{1,,1,,2}2,,2}2{1,,1{1,,2,,2}2,,2}2,,1,,2} has recursion level C(C(Ω22+C(Ω2+C(C(Ω2+C(Ω22,0),C(Ω22,0)),C(C(Ω22,C(Ω22,0)),C(Ω22,0))),0),0),0)
  • {1{1,,1{1{1,,1{1,,1,,2}2,,2}2,,1,,2}2,,2}2{1,,1{1,,2,,2}2,,2}2,,1,,2} has recursion level C(C(Ω22+C(Ω2+C(C(Ω2+C(C(Ω2+C(Ω22,0),C(Ω22,0)),C(Ω22,0)),C(Ω22,0)),C(C(Ω22,C(Ω22,0)),C(Ω22,0))),0),0),0)
  • {1{1,,1{1{1,,1{1,,2,,2}2,,2}2,,1,,2}2,,2}2{1,,1{1,,2,,2}2,,2}2,,1,,2} has recursion level C(C(Ω22+C(Ω2+C(C(Ω2+C(C(Ω22,C(Ω22,0)),C(Ω22,0)),C(Ω22,0)),C(C(Ω22,C(Ω22,0)),C(Ω22,0))),0),0),0)
  • {1{1,,1{1{1,,1{1,,1,,2}2,,2}2{1,,1{1,,2,,2}2,,2}2,,1,,2}2,,2}2{1,,1{1,,2,,2}2,,2}2,,1,,2} has recursion level C(C(Ω22+C(Ω2+C(C(Ω2+C(C(Ω2+C(Ω22,0),C(Ω22,0)),C(C(Ω22,C(Ω22,0)),C(Ω22,0))),C(Ω22,0)),C(C(Ω22,C(Ω22,0)),C(Ω22,0))),0),0),0)
  • {1{1,,1{1,,2,,2}2,,2}3,,1,,2} has recursion level C(C(Ω22+C(Ω2+C(C(Ω22,C(Ω22,0)),C(C(Ω22,C(Ω22,0)),C(Ω22,0))),0),0),0)
  • {1{1,,1{1,,2,,2}2,,2}1,2,,1,,2} has recursion level C(C(Ω22+C(Ω2+C(C(Ω22,C(Ω22,0))+1,C(Ω22,0)),0),0),0)
  • {1{1,,1{1,,2,,2}2,,2}1{1,,1,,2}2,,1,,2} has recursion level C(C(Ω22+C(Ω2+C(C(Ω22,C(Ω22,0))+C(Ω22,0),C(Ω22,0)),0),0),0)
  • {1{1,,1{1,,2,,2}2,,2}1{1,,2,,2}2,,1,,2} has recursion level C(C(Ω22+C(Ω2+C(C(Ω22,C(Ω22,0))+C(Ω2,C(Ω22,0)),C(Ω22,0)),0),0),0)
  • {1{1,,1{1,,2,,2}2,,2}1{1,,3,,2}2,,1,,2} has recursion level C(C(Ω22+C(Ω2+C(C(Ω22,C(Ω22,0))+C(C(Ω2,C(Ω2,C(Ω22,0))),C(Ω2,C(Ω22,0))),C(Ω22,0)),0),0),0)
  • {1{1,,1{1,,2,,2}2,,2}1{1,,1{1,,1,,2}2,,2}2,,1,,2} has recursion level C(C(Ω22+C(Ω2+C(C(Ω22,C(Ω22,0))+C(C(Ω2+C(Ω22,0),C(Ω22,0)),C(Ω2,C(Ω22,0))),C(Ω22,0)),0),0),0)
  • {1{1,,1{1,,2,,2}2,,2}1{1,,1{1,,2,,2}2,,2}1,2,,1,,2} has recursion level C(C(Ω22+C(Ω2+C(C(Ω22,C(Ω22,0))+C(C(Ω22,C(Ω22,0)),C(Ω2,C(Ω22,0)))+1,C(Ω22,0)),0),0),0)
  • {1{2,,1{1,,2,,2}2,,2}2,,1,,2} has recursion level C(C(Ω22+C(Ω2+C(C(Ω22,C(Ω22,0))+ω^(C(C(Ω22,C(Ω22,0)),C(Ω2,C(Ω22,0)))+1),C(Ω22,0)),0),0),0)
  • {1{1{1,,2,,2}2,,1{1,,2,,2}2,,2}2,,1,,2} has recursion level C(C(Ω22+C(Ω2+C(C(Ω22,C(Ω22,0))+ω^(C(C(Ω22,C(Ω22,0)),C(Ω2,C(Ω22,0)))+C(Ω2,C(Ω22,0))),C(Ω22,0)),0),0),0)

So {1{1,,2,,2}2,,1{1,,2,,2}2,,2} has recursion level C(C(Ω22+C(Ω2+C(C(Ω22,C(Ω22,0))+ω^(C(C(Ω22,C(Ω22,0)),C(Ω2,C(Ω22,0)))+C(Ω2,C(Ω22,0))),C(Ω22,0)),0),0),0).

  • {1{2{1,,1{1,,2,,2}2,,2}2,,2,,2}2,,1,,2} has recursion level C(C(Ω22+C(Ω2+C(C(Ω22,C(Ω22,0))+ω^(C(C(Ω22,C(Ω22,0)),C(Ω2,C(Ω22,0)))+1),C(Ω22,0)),0),0),0)
  • {1{1,,3,,2}2{1,,1{1,,2,,2}2,,2}2,,2,,2} has recursion level C(C(Ω22+C(Ω2+C(C(Ω22,C(Ω22,0))+C(C(Ω2,C(Ω2,C(Ω22,0))),C(C(Ω22,C(Ω22,0)),C(Ω2,C(Ω22,0)))),C(Ω22,0)),0),0),0)
  • {1{1,,1{1,,1,,2}2,,2}2{1,,1{1,,2,,2}2,,2}2,,2,,2} has recursion level C(C(Ω22+C(Ω2+C(C(Ω22,C(Ω22,0))+C(C(Ω2+C(Ω22,0),C(Ω22,0)),C(C(Ω22,C(Ω22,0)),C(Ω2,C(Ω22,0)))),C(Ω22,0)),0),0),0)
  • {1{1,,1{1{1,,1{1,,2,,2}2,,2}1{1,,2,,2}2,,1,,2}2,,2}2{1,,1{1,,2,,2}2,,2}2,,2,,2} has recursion level C(C(Ω22+C(Ω2+C(C(Ω22,C(Ω22,0))+C(C(Ω2+C(C(Ω22,C(Ω22,0))+C(Ω2,C(Ω22,0)),C(Ω22,0)),C(Ω22,0)),C(C(Ω22,C(Ω22,0)),C(Ω2,C(Ω22,0)))),C(Ω22,0)),0),0),0)
  • {1{1,,1{1{1{1,,1{1,,1,,2}2,,2}2{1,,1{1,,2,,2}2,,2}2,,2,,2}2,,1,,2}2,,2}2{1,,1{1,,2,,2}2,,2}2,,2,,2} has recursion level C(C(Ω22+C(Ω2+C(C(Ω22,C(Ω22,0))+C(C(Ω2+C(C(Ω22,C(Ω22,0))+C(C(Ω2+C(Ω22,0),C(Ω22,0)),C(C(Ω22,C(Ω22,0)),C(Ω2,C(Ω22,0)))),C(Ω22,0)),C(Ω22,0)),C(C(Ω22,C(Ω22,0)),C(Ω2,C(Ω22,0)))),C(Ω22,0)),0),0),0)
  • {1{1,,1{1,,2,,2}2,,2}3,,2,,2} has recursion level C(C(Ω22+C(Ω2+C(C(Ω22,C(Ω22,0))+C(C(Ω22,C(Ω22,0)),C(C(Ω22,C(Ω22,0)),C(Ω2,C(Ω22,0)))),C(Ω22,0)),0),0),0)
  • {1{1,,1{1,,2,,2}2,,2}1,2,,2,,2} has recursion level C(C(Ω22+C(Ω2+C(C(Ω22,C(Ω22,0))+C(C(Ω22,C(Ω22,0))+1,C(Ω2,C(Ω22,0))),C(Ω22,0)),0),0),0)
  • {1{1,,1{1,,2,,2}2,,2}1{1,,2,,2}2,,2,,2} has recursion level C(C(Ω22+C(Ω2+C(C(Ω22,C(Ω22,0))+C(C(Ω22,C(Ω22,0))+C(Ω2,C(Ω22,0)),C(Ω2,C(Ω22,0))),C(Ω22,0)),0),0),0)
  • {1{1,,1{1,,2,,2}2,,2}1{1,,3,,2}2,,2,,2} has recursion level C(C(Ω22+C(Ω2+C(C(Ω22,C(Ω22,0))+C(Ω2,C(Ω2,C(Ω22,0))),C(Ω22,0)),0),0),0)
  • {1{1,,1{1,,2,,2}2,,2}1{1,,1{1,,2,,2}2,,2}2,,2,,2} has recursion level C(C(Ω22+C(Ω2+C(C(Ω22,C(Ω22,0))+C(C(Ω22,C(Ω22,0)),C(Ω2,C(Ω2,C(Ω22,0)))),C(Ω22,0)),0),0),0)
  • {1{2,,1{1,,2,,2}2,,2}2,,2,,2} has recursion level C(C(Ω22+C(Ω2+C(C(Ω22,C(Ω22,0))+ω^(C(C(Ω22,C(Ω22,0)),C(Ω2,C(Ω2,C(Ω22,0))))+1),C(Ω22,0)),0),0),0)
  • {1{1{1,,3,,2}2,,1{1,,2,,2}2,,2}2,,2,,2} has recursion level C(C(Ω22+C(Ω2+C(C(Ω22,C(Ω22,0))+ω^(C(C(Ω22,C(Ω22,0)),C(Ω2,C(Ω2,C(Ω22,0))))+C(Ω2,C(Ω2,C(Ω22,0)))),C(Ω22,0)),0),0),0)

So {1{1,,3,,2}2,,1{1,,2,,2}2,,2} has recursion level C(C(Ω22+C(Ω2+C(C(Ω22,C(Ω22,0))+ω^(C(C(Ω22,C(Ω22,0)),C(Ω2,C(Ω2,C(Ω22,0))))+C(Ω2,C(Ω2,C(Ω22,0)))),C(Ω22,0)),0),0),0). And further, the recursion level of {1{1,,k,,2}2,,1{1,,2,,2}2,,2} is between C(C(Ω22+C(Ω2+C(C(Ω22,C(Ω22,0))+C(Ω2,C(Ω2,…C(Ω2,C(Ω22,0)))),C(Ω22,0)),0),0),0) (with k-1 Ω2′s in the blue part) and C(C(Ω22+C(Ω2+C(C(Ω22,C(Ω22,0))+C(Ω2,C(Ω2,…C(Ω2,C(Ω22,0)))),C(Ω22,0)),0),0),0) (with k Ω2′s in the blue part), so {1{1,,1,2,,2}2,,1{1,,2,,2}2,,2} has recursion level C(C(Ω22+C(Ω2+C(C(Ω22,C(Ω22,0))+C(Ω2+1,C(Ω22,0)),C(Ω22,0)),0),0),0).

  • {1{1,,1{1,,1,,2}2,,2}2,,1{1,,2,,2}2,,2} has recursion level C(C(Ω22+C(Ω2+C(C(Ω22,C(Ω22,0))+C(Ω2+C(Ω22,0),C(Ω22,0)),C(Ω22,0)),0),0),0)
  • {1{1,,1{1{1,,1{1,,2,,2}2,,2}2,,1,,2}2,,2}2,,1{1,,2,,2}2,,2} has recursion level C(C(Ω22+C(Ω2+C(C(Ω22,C(Ω22,0))+C(Ω2+C(C(Ω22,C(Ω22,0)),C(Ω22,0)),C(Ω22,0)),C(Ω22,0)),0),0),0)
  • {1{1,,1{1{1{1,,1{1,,1,,2}2,,2}2,,1{1,,2,,2}2,,2}2,,1,,2}2,,2}2,,1{1,,2,,2}2,,2} has recursion level C(C(Ω22+C(Ω2+C(C(Ω22,C(Ω22,0))+C(Ω2+C(C(Ω22,C(Ω22,0))+C(Ω2+C(Ω22,0),C(Ω22,0)),C(Ω22,0)),C(Ω22,0)),C(Ω22,0)),0),0),0)
  • {1{1,,1{1,,2,,2}2,,2}2,,1{1,,2,,2}2,,2} has recursion level C(C(Ω22+C(Ω2+C(C(Ω22,C(Ω22,0))2,C(Ω22,0)),0),0),0)
  • {1{1,,1{1,,2,,2}2,,2}1{1,,1{1,,2,,2}2,,2}2,,1{1,,2,,2}2,,2} has recursion level C(C(Ω22+C(Ω2+C(ω^(C(Ω22,C(Ω22,0))2),C(Ω22,0)),0),0),0)
  • {1{1{1,,1{1,,2,,2}2,,2}2,,1{1,,2,,2}2,,2}2,,1{1,,2,,2}2,,2} has recursion level C(C(Ω22+C(Ω2+C(ω^ω^(C(Ω22,C(Ω22,0))2),C(Ω22,0)),0),0),0)
  • {1,,2{1,,2,,2}2,,2} has recursion level C(C(Ω22+C(Ω2+C(C(Ω2,C(Ω22,C(Ω22,0))),C(Ω22,0)),0),0),0)
  • {1,,1,2{1,,2,,2}2,,2} has recursion level C(C(Ω22+C(Ω2+C(C(Ω2+1,C(Ω22,C(Ω22,0))),C(Ω22,0)),0),0),0)
  • {1,,1{1,,1,,2}2{1,,2,,2}2,,2} has recursion level C(C(Ω22+C(Ω2+C(C(Ω2+C(Ω22,0),C(Ω22,C(Ω22,0))),C(Ω22,0)),0),0),0)
  • {1,,1{1{1,,1{1,,2,,2}2,,2}2,,1,,2}2{1,,2,,2}2,,2} has recursion level C(C(Ω22+C(Ω2+C(C(Ω2+C(C(Ω22,C(Ω22,0)),C(Ω22,0)),C(Ω22,C(Ω22,0))),C(Ω22,0)),0),0),0)
  • {1,,1{1{1,,1{1,,1,,2}2{1,,2,,2}2,,2}2,,1,,2}2{1,,2,,2}2,,2} has recursion level C(C(Ω22+C(Ω2+C(C(Ω2+C(C(Ω2+C(Ω22,0),C(Ω22,C(Ω22,0))),C(Ω22,0)),C(Ω22,C(Ω22,0))),C(Ω22,0)),0),0),0)
  • {1,,1{1,,2,,2}3,,2} has recursion level C(C(Ω22+C(Ω2+C(C(Ω22,C(Ω22,C(Ω22,0))),C(Ω22,0)),0),0),0)
  • {1,,1{1,,2,,2}1,2,,2} has recursion level C(C(Ω22+C(Ω2+C(C(Ω22+1,0),C(Ω22,0)),0),0),0)
  • {1,,1{1,,2,,2}1{1,,1{1,,1,,2}2}2,,2} has recursion level C(C(Ω22+C(Ω2+C(C(Ω22+C(Ω2+C(Ω22,0),0),0),C(Ω22,0)),0),0),0)
  • {1,,1{1,,2,,2}1{1,,1{1,,1{1,,2,,2}1,2,,2}2}2,,2} has recursion level C(C(Ω22+C(Ω2+C(C(Ω22+C(Ω2+C(C(Ω22+1,0),C(Ω22,0)),0),0),C(Ω22,0)),0),0),0)
  • {1,,1{1,,2,,2}1{1,,1,,2}2,,2} has recursion level C(C(Ω22+C(Ω2+C(Ω2,C(Ω22,0)),0),0),0)

Now let separator ◊ = {1,,1{1,,2,,2}1{1,,1,,2}2,,2} and let ordinal a = C(Ω2+C(Ω22,0),0), and b = C(Ω2+C(Ω2,C(Ω22,0)),0).

  • {1,,1{1{1,,1◊2}2,,1{1,,1,,2}2}2{1,,1,,2}2} has recursion level C(C(Ω22+b,0),0)
  • {1,,1{1,,1{1,,1,,2}2}2{1{1,,1◊2}2,,1{1,,1,,2}2}2{1,,1,,2}2} has recursion level C(C(Ω22,C(Ω22+b,0)),0)
  • {1,,1{2,,1{1,,1,,2}2}2{1{1,,1◊2}2,,1{1,,1,,2}2}2{1,,1,,2}2} has recursion level C(C(Ω22+ω^(a+1),C(Ω22+b,0)),0)
  • {1,,1{1{1,,1{1,,1,,2}1,2}2,,1{1,,1,,2}2}2{1{1,,1◊2}2,,1{1,,1,,2}2}2{1,,1,,2}2} has recursion level C(C(Ω22+C(Ω2+C(Ω22,0)+1,0),C(Ω22+b,0)),0)
  • {1,,1{1{1,,1{1,,2,,2}2}2,,1{1,,1,,2}2}2{1{1,,1◊2}2,,1{1,,1,,2}2}2{1,,1,,2}2} has recursion level C(C(Ω22+C(Ω2+C(C(Ω2,C(Ω22,0)),C(Ω22,0)),0),C(Ω22+b,0)),0)
  • {1,,1{1{1,,1{1,,1{1,,2,,2}2,,2}2}2,,1{1,,1,,2}2}2 {1{1,,1◊2}2,,1{1,,1,,2}2}2{1,,1,,2}2} has recursion level C(C(Ω22+C(Ω2+C(C(Ω22,C(Ω22,0)),C(Ω22,0)),0),C(Ω22+b,0)),0)
  • {1,,1{1{1,,1{1,,1{1,,2,,2}1,2,,2}2}2,,1{1,,1,,2}2}2{1{1,,1◊2}2,,1{1,,1,,2}2}2{1,,1,,2}2} has recursion level C(C(Ω22+C(Ω2+C(C(Ω22+1,0),C(Ω22,0)),0),C(Ω22+b,0)),0)
  • {1,,1{1{1,,1{1,,1{1,,2,,2}1{1,,1{1,,1,,2}2}2,,2}2}2,,1{1,,1,,2}2}2{1{1,,1◊2}2,,1{1,,1,,2}2}2{1,,1,,2}2} has recursion level C(C(Ω22+C(Ω2+C(C(Ω22+a,0),C(Ω22,0)),0),C(Ω22+b,0)),0)
  • {1,,1{1{1,,1{1,,1{1,,2,,2}1{1,,1{1,,1{1,,2,,2}1,2,,2}2}2,,2}2}2,,1{1,,1,,2}2}2 {1{1,,1◊2}2,,1{1,,1,,2}2}2{1,,1,,2}2} has recursion level C(C(Ω22+C(Ω2+C(C(Ω22+C(Ω2+C(C(Ω22+1,0),C(Ω22,0)),0),0),C(Ω22,0)),0),C(Ω22+b,0)),0)
  • {1,,1{1{1,,1◊2}2,,1{1,,1,,2}2}3{1,,1,,2}2} has recursion level C(C(Ω22+b,C(Ω22+b,0)),0)
  • {1,,1{1{1,,1◊2}2,,1{1,,1,,2}2}1,2{1,,1,,2}2} has recursion level C(C(Ω22+b+1,0),0)
  • {1,,1{1{1,,1◊2}2,,1{1,,1,,2}2}1{1,,1{1,,1,,2}2}2{1,,1,,2}2} has recursion level C(C(Ω22+b+a,0),0)
  • {1,,1{1{1,,1◊2}2,,1{1,,1,,2}2}1{1{1,,1{1,,1,,2}3}2,,1{1,,1,,2}2}2{1,,1,,2}2} has recursion level C(C(Ω22+b+C(C(Ω22,0),a),0),0)
  • {1,,1{1{1,,1◊2}2,,1{1,,1,,2}2}1{1{1,,1◊2}2,,1{1,,1,,2}2}2{1,,1,,2}2} has recursion level C(C(Ω22+b+C(C(Ω22+b,0),a),0),0)
  • {1,,1{2{1,,1◊2}2,,1{1,,1,,2}2}2{1,,1,,2}2} has recursion level C(C(Ω22+b+ω^(C(C(Ω22+b,0),a)+1),0),0)
  • {1,,1{1{1,,1◊2}1,2,,1{1,,1,,2}2}2{1,,1,,2}2} has recursion level C(C(Ω22+b+C(C(Ω22+b,0)+1,a),0),0)
  • {1,,1{1{1{1,,1,2{1,,1,,2}2}2,,1◊2}2,,1{1,,1,,2}2}2{1,,1,,2}2} has recursion level C(C(Ω22+b+C(C(Ω22+b,0)+C(Ω2+1,a),a),0),0)
  • {1{1,,1{1,,1,,2}3}2,,1◊2} has recursion level C(C(Ω22+b+C(C(Ω22+b,0)+C(Ω22,0),a),0),0)
  • {1{1,,1{1,,1,,2}1,2}2,,1◊2} has recursion level C(C(Ω22+b+C(C(Ω22+b,0)+C(Ω22+C(Ω2+C(Ω22,0)+1,0),0),a),0),0)
  • {1{1,,1◊2}2,,1◊2} has recursion level C(C(Ω22+b+C(C(Ω22+b,0)2,a),0),0)
  • {1,,2◊2} has recursion level C(C(Ω22+b+C(C(Ω2,C(Ω22+b,0)),a),0),0)
  • {1,,1{1,,1{1,,1,,2}2}2◊2} has recursion level C(C(Ω22+b+C(C(Ω2+a,C(Ω22+b,0)),a),0),0)
  • {1,,1{1,,1{1,,1,,2}3}2◊2} has recursion level C(C(Ω22+b+C(C(Ω22,C(Ω22+b,0)),a),0),0)
  • {1,,1{1,,1◊2}2◊2} has recursion level C(C(Ω22+b+C(C(Ω22+b,C(Ω22+b,0)),a),0),0)
  • {1,,1{1,,1◊2}1,2◊2} has recursion level C(C(Ω22+b+C(C(Ω22+b+1,0),a),0),0)
  • {1,,1{1,,1◊2}1{1,,1{1,,1,,2}2}2◊2} has recursion level C(C(Ω22+b+C(C(Ω22+b+a,0),a),0),0)
  • {1,,1{1,,1◊2}1{1,,2{1,,1,,2}2}2◊2} has recursion level C(C(Ω22+b+C(Ω2,a),0),0)
  • {1,,1{1,,1◊2}1{1,,1{1,,1,,2}3}2◊2} has recursion level C(C(Ω22+b+C(Ω2+C(Ω22,0),a),0),0)
  • {1,,1{1,,1◊2}1{1,,1{1,,1,,2}1,2}2◊2} has recursion level C(C(Ω22+b+C(Ω2+C(Ω22,0)+1,0),0),0)
  • {1,,1{1,,1◊2}1{1,,1{1,,1{1,,2,,2}1,2,,2}2}2◊2} has recursion level C(C(Ω22+b+C(Ω2+C(C(Ω22+1,0),C(Ω22,0)),0),0),0)
  • {1,,1{1,,1◊2}1{1,,1{1,,1{1,,2,,2}1{1,,1{1,,1{1,,2,,2}1,2,,2}2}2,,2}2}2◊2} has recursion level C(C(Ω22+b+C(Ω2+C(C(Ω22+C(Ω2+C(C(Ω22+1,0),C(Ω22,0)),0),0),C(Ω22,0)),0),0),0)
  • {1,,1{1,,1◊2}1{1,,1◊2}2◊2} has recursion level C(C(Ω22+b+C(Ω2+C(C(Ω22+b,0),C(Ω22,0)),0),0),0)
  • {1,,1{2,,1◊2}2◊2} has recursion level C(C(Ω22+b+ω^(C(Ω2+C(C(Ω22+b,0),C(Ω22,0)),0)+1),0),0)

Now let separator ◊ = {1,,1{1,,2,,2}1{1,,1,,2}2,,2} and let ordinal b = C(Ω2+C(Ω2,C(Ω22,0)),0), and a = C(Ω2+C(C(Ω22+b,0),C(Ω22,0)),0) = C(Ω2+C(C(Ω22+C(Ω2+C(Ω2,C(Ω22,0)),0),0),C(Ω22,0)),0).

  • {1,,1{1{1,,1◊2}2,,1◊2}2◊2} has recursion level C(C(Ω22+b+ω^(a2),0),0)
  • {1,,1{1{1,,2◊2}2,,1◊2}2◊2} has recursion level C(C(Ω22+b+C(C(Ω2,a),a),0),0)
  • {1,,1{1{1,,1,2◊2}2,,1◊2}2◊2} has recursion level C(C(Ω22+b+C(C(Ω2+1,a),a),0),0)
  • {1,,1{1{1,,1{1,,1{1,,1{1,,2,,2}1,2,,2}2}2◊2}2,,1◊2}2◊2} has recursion level C(C(Ω22+b+C(C(Ω2+C(Ω2+C(C(Ω22+1,0),C(Ω22,0)),0),a),a),0),0)
  • {1,,1{1{1,,1{1,,1◊2}2◊2}2,,1◊2}2◊2} has recursion level C(C(Ω22+b+C(C(Ω2+a,a),a),0),0)
  • {1,,1{1,,2◊2}2◊2} has recursion level C(C(Ω22+b+C(C(Ω2+C(Ω2,a),a),a),0),0)
  • {1,,1{1,,1{1,,2◊2}2◊2}2◊2} has recursion level C(C(Ω22+b+C(C(Ω2+C(Ω2+C(Ω2,a),a),a),a),0),0)
  • {1,,1{1,,1,,2}2◊2} has recursion level C(C(Ω22+b+C(C(Ω22,0),a),0),0)
  • {1,,1{1{1,,1{1,,1,,2}2◊2}2,,1{1,,1,,2}2}1,2{1,,1,,2}2} has recursion level C(C(Ω22+b+C(C(Ω22,0),a)+1,0),0)
  • {1,,1{1{1,,1{1,,1,,2}2◊2}2,,1{1,,1,,2}2}1{1,,1{1,,1,,2}2}2{1,,1,,2}2} has recursion level C(C(Ω22+b+C(C(Ω22,0),a)+C(Ω2+C(Ω22,0),0),0),0)
  • {1,,1{1{1,,1{1,,1,,2}2◊2}2,,1{1,,1,,2}2}1{1{1,,1{1,,1,,2}2◊2}2,,1{1,,1,,2}2}2{1,,1,,2}2} has recursion level C(C(Ω22+b+C(C(Ω22,0),a)+C(C(Ω22+b+C(C(Ω22,0),a),0),C(Ω2+C(Ω22,0),0)),0),0)
  • {1,,1{2{1,,1{1,,1,,2}2◊2}2,,1{1,,1,,2}2}2{1,,1,,2}2} has recursion level C(C(Ω22+b+C(C(Ω22,0),a)+ω^(C(C(Ω22+b+C(C(Ω22,0),a),0),C(Ω2+C(Ω22,0),0))+1),0),0)
  • {1,,1{1{1,,1{1,,1,,2}2◊2}1,2,,1{1,,1,,2}2}2{1,,1,,2}2} has recursion level C(C(Ω22+b+C(C(Ω22,0),a)+C(C(Ω22+b+C(C(Ω22,0),a),0)+1,C(Ω2+C(Ω22,0),0)),0),0)
  • {1{1,,1{1,,1,,2}3}2,,1{1,,1,,2}2◊2} has recursion level C(C(Ω22+b+C(C(Ω22,0),a)+C(C(Ω22+b+C(C(Ω22,0),a),0)+C(Ω22,0),C(Ω2+C(Ω22,0),0)),0),0)
  • {1{1,,1◊2}2,,1{1,,1,,2}2◊2} has recursion level C(C(Ω22+b+C(C(Ω22,0),a)+C(C(Ω22+b+C(C(Ω22,0),a),0)+C(Ω22+b,0),C(Ω2+C(Ω22,0),0)),0),0)
  • {1{1{1,,2◊2}2,,1◊2}2,,1{1,,1,,2}2◊2} has recursion level C(C(Ω22+b+C(C(Ω22,0),a)+C(C(Ω22+b+C(C(Ω22,0),a),0)+C(C(Ω2,C(Ω22+b,0)),C(Ω22+b,0)),C(Ω2+C(Ω22,0),0)),0),0)
  • {1{1{1,,1{1,,1,,2}2◊2}2,,1◊2}2,,1{1,,1,,2}2◊2} has recursion level C(C(Ω22+b+C(C(Ω22,0),a)+C(C(Ω22+b+C(C(Ω22,0),a),0)+C(C(Ω22+b+C(C(Ω22,0),a),0),C(Ω22+b,0)),C(Ω2+C(Ω22,0),0)),0),0)
  • {1{1,,2◊2}2,,1{1,,1,,2}2◊2} has recursion level C(C(Ω22+b+C(C(Ω22,0),a)+C(C(Ω22+b+C(C(Ω22,0),a),0)+C(Ω2,C(Ω22+b,0)),C(Ω2+C(Ω22,0),0)),0),0)
  • {1{1,,1{1,,1◊2}1,2◊2}2,,1{1,,1,,2}2◊2} has recursion level C(C(Ω22+b+C(C(Ω22,0),a)+C(C(Ω22+b+C(C(Ω22,0),a),0)+C(Ω22+b+1,0),C(Ω2+C(Ω22,0),0)),0),0)
  • {1{1,,1{1,,1,,2}2◊2}2,,1{1,,1,,2}2◊2} has recursion level C(C(Ω22+b+C(C(Ω22,0),a)+C(C(Ω22+b+C(C(Ω22,0),a),0)2,C(Ω2+C(Ω22,0),0)),0),0)
  • {1,,2{1,,1,,2}2◊2} has recursion level C(C(Ω22+b+C(C(Ω22,0),a)+C(C(Ω2,C(Ω22+b+C(C(Ω22,0),a),0)),C(Ω2+C(Ω22,0),0)),0),0)
  • {1,,1{1,,1{1,,1,,2}2◊2}1,2{1,,1,,2}2◊2} has recursion level C(C(Ω22+b+C(C(Ω22,0),a)+C(C(Ω22+b+C(C(Ω22,0),a)+1,0),C(Ω2+C(Ω22,0),0)),0),0)
  • {1,,1{1,,1{1,,1,,2}2◊2}1{1,,1{1,,1,,2}2}2{1,,1,,2}2◊2} has recursion level C(C(Ω22+b+C(C(Ω22,0),a)+C(C(Ω22+b+C(C(Ω22,0),a)+C(Ω2+C(Ω22,0),0),0),C(Ω2+C(Ω22,0),0)),0),0)
  • {1,,1{1,,1{1,,1,,2}2◊2}1{1,,2{1,,1,,2}2}2{1,,1,,2}2◊2} has recursion level C(C(Ω22+b+C(C(Ω22,0),a)+C(Ω2,C(Ω2+C(Ω22,0),0)),0),0)
  • {1,,1{1,,1{1,,1,,2}2◊2}1{1,,1{1,,1,,2}1,2}2{1,,1,,2}2◊2} has recursion level C(C(Ω22+b+C(C(Ω22,0),a)+C(Ω2+C(Ω22,0)+1,0),0),0)
  • {1,,1{1,,1{1,,1,,2}2◊2}1{1,,1◊2}2{1,,1,,2}2◊2} has recursion level C(C(Ω22+b+C(C(Ω22,0),a)+a,0),0)
  • {1,,1{1,,1{1,,1,,2}2◊2}1{1{1,,2◊2}2,,1◊2}2{1,,1,,2}2◊2} has recursion level C(C(Ω22+b+C(C(Ω22,0),a)+C(C(Ω2,a),a),0),0)
  • {1,,1{1,,1{1,,1,,2}2◊2}1{1{1,,1{1,,1◊2}2◊2}2,,1◊2}2{1,,1,,2}2◊2} has recursion level C(C(Ω22+b+C(C(Ω22,0),a)+C(C(Ω2+a,a),a),0),0)
  • {1,,1{1,,1{1,,1,,2}2◊2}1{1{1,,1{1,,1,,2}2◊2}2,,1◊2}2{1,,1,,2}2◊2} has recursion level C(C(Ω22+b+C(C(Ω22,0),a)2,0),0)
  • {1,,1{1,,1{1,,1,,2}2◊2}1{2{1,,1{1,,1,,2}2◊2}2,,1◊2}2{1,,1,,2}2◊2} has recursion level C(C(Ω22+b+ω^(C(C(Ω22,0),a)+1),0),0)
  • {1,,1{1,,1{1,,1,,2}2◊2}1{1{1,,1{1,,1,,2}2◊2}1,2,,1◊2}2{1,,1,,2}2◊2} has recursion level C(C(Ω22+b+C(C(Ω22,0)+1,a),0),0)
  • {1,,1{1,,1{1,,1,,2}2◊2}1{1{1{1,,1{1,,1,,2}2◊2}2,,1{1,,1,,2}2◊2}2,,1◊2}2{1,,1,,2}2◊2} has recursion level C(C(Ω22+b+C(C(Ω22,0)2,a),0),0)
  • {1,,1{1,,1{1,,1,,2}2◊2}1{1{1,,2{1,,1,,2}2◊2}2,,1◊2}2{1,,1,,2}2◊2} has recursion level C(C(Ω22+b+C(C(Ω2,C(Ω22,0)),a),0),0)
  • {1,,1{1,,1{1,,1,,2}2◊2}1{1{1,,1{1,,1{1,,1,,2}2◊2}2{1,,1,,2}2◊2}2,,1◊2}2{1,,1,,2}2◊2} has recursion level C(C(Ω22+b+C(C(Ω22,C(Ω22,0)),a),0),0)
  • {1,,1{1,,1{1,,1,,2}2◊2}1{1{1,,1{1,,1{1,,1,,2}2◊2}1,2{1,,1,,2}2◊2}2,,1◊2}2{1,,1,,2}2◊2} has recursion level C(C(Ω22+b+C(C(Ω22+1,0),a),0),0)
  • {1,,1{1,,1{1,,1,,2}2◊2}1{1{1,,1{1,,1{1,,1,,2}2◊2}1{1,,1◊2}2{1,,1,,2}2◊2}2,,1◊2}2{1,,1,,2}2◊2} has recursion level C(C(Ω22+b+C(C(Ω22+a,0),a),0),0)
  • {1,,1{1,,1{1,,1,,2}2◊2}1{1,,2◊2}2{1,,1,,2}2◊2} has recursion level C(C(Ω22+b+C(C(Ω22+C(Ω2,a),0),a),0),0) (it’s different to previous comparisons on {1,,1{1,,1{1,,1,,2}k+1}1{1,,2{1,,1,,2}k}2{1,,1,,2}k+1}′s)
  • {1,,1{1,,1{1,,1,,2}2◊2}1{1,,1{1,,2◊2}2◊2}2{1,,1,,2}2◊2} has recursion level C(C(Ω22+b+C(C(Ω22+C(Ω2+C(Ω2,a),a),0),a),0),0)
  • {1,,1{1,,1{1,,1,,2}2◊2}1{1,,1{1,,1,,2}2◊2}2{1,,1,,2}2◊2} has recursion level C(C(Ω22+b+C(C(Ω22+C(Ω2+C(Ω22,0),a),0),a),0),0)
  • {1,,1{1{1,,1{1,,1{1,,1,,2}2◊2}1{1,,1{1,,1,,2}2◊2}2{1,,1,,2}2◊2}2,,1{1,,1,,2}2}1,2{1,,1,,2}2} has recursion level C(C(Ω22+b+C(C(Ω22+C(Ω2+C(Ω22,0),a),0),a)+1,0),0)
  • {1,,1{1{1,,1{1,,1{1,,1,,2}2◊2}1{1,,1{1,,1,,2}2◊2}2{1,,1,,2}2◊2}2,,1{1,,1,,2}2} 1{1,,1{1,,1,,2}2}2{1,,1,,2}2} has recursion level C(C(Ω22+b+C(C(Ω22+C(Ω2+C(Ω22,0),a),0),a)+C(Ω2+C(Ω22,0),0),0),0)
  • {1,,1{1{1,,1{1,,1{1,,1,,2}2◊2}1{1,,1{1,,1,,2}2◊2}2{1,,1,,2}2◊2}2,,1{1,,1,,2}2} 1{1{1,,1{1,,1{1,,1,,2}2◊2}1{1,,1{1,,1,,2}2◊2}2{1,,1,,2}2◊2}2,,1{1,,1,,2}2}2{1,,1,,2}2} has recursion level C(C(Ω22+b+C(C(Ω22+C(Ω2+C(Ω22,0),a),0),a)+C(C(Ω22+b+C(C(Ω22+C(Ω2+C(Ω22,0),a),0),a),0),C(Ω2+C(Ω22,0),0)),0),0)
  • {1,,1{1{1,,1{1,,1{1,,1,,2}2◊2}1{1,,1{1,,1,,2}2◊2}2{1,,1,,2}2◊2}1,2,,1{1,,1,,2}2}2{1,,1,,2}2} has recursion level C(C(Ω22+b+C(C(Ω22+C(Ω2+C(Ω22,0),a),0),a)+C(C(Ω22+b+C(C(Ω22+C(Ω2+C(Ω22,0),a),0),a),0)+1,C(Ω2+C(Ω22,0),0)),0),0)
  • {1{1,,1{1,,1,,2}3}2,,1{1,,1{1,,1,,2}2◊2}1{1,,1{1,,1,,2}2◊2}2{1,,1,,2}2◊2} has recursion level C(C(Ω22+b+C(C(Ω22+C(Ω2+C(Ω22,0),a),0),a)+C(C(Ω22+b+C(C(Ω22+C(Ω2+C(Ω22,0),a),0),a),0)+C(Ω22,0),C(Ω2+C(Ω22,0),0)),0),0)
  • {1{1,,1◊2}2,,1{1,,1{1,,1,,2}2◊2}1{1,,1{1,,1,,2}2◊2}2{1,,1,,2}2◊2} has recursion level C(C(Ω22+b+C(C(Ω22+C(Ω2+C(Ω22,0),a),0),a)+C(C(Ω22+b+C(C(Ω22+C(Ω2+C(Ω22,0),a),0),a),0)+C(Ω22+b,0),C(Ω2+C(Ω22,0),0)),0),0)
  • {1{1,,1{1,,1{1,,1,,2}2◊2}1{1,,1{1,,1,,2}2◊2}2{1,,1,,2}2◊2} 2,,1{1,,1{1,,1,,2}2◊2}1{1,,1{1,,1,,2}2◊2}2{1,,1,,2}2◊2} has recursion level C(C(Ω22+b+C(C(Ω22+C(Ω2+C(Ω22,0),a),0),a)+C(C(Ω22+b+C(C(Ω22+C(Ω2+C(Ω22,0),a),0),a),0)2,C(Ω2+C(Ω22,0),0)),0),0)
  • {1,,2{1,,1{1,,1,,2}2◊2}1{1,,1{1,,1,,2}2◊2}2{1,,1,,2}2◊2} has recursion level C(C(Ω22+b+C(C(Ω22+C(Ω2+C(Ω22,0),a),0),a)+C(C(Ω2,C(Ω22+b+C(C(Ω22+C(Ω2+C(Ω22,0),a),0),a),0)),C(Ω2+C(Ω22,0),0)),0),0)
  • {1,,1{1,,1{1,,1,,2}2◊2}1{1,,1{1,,1,,2}2◊2}3{1,,1,,2}2◊2} has recursion level C(C(Ω22+b+C(C(Ω22+C(Ω2+C(Ω22,0),a),0),a)+C(C(Ω22+b+C(C(Ω22+C(Ω2+C(Ω22,0),a),0),a),C(Ω22+b+C(C(Ω22+C(Ω2+C(Ω22,0),a),0),a),0)),C(Ω2+C(Ω22,0),0)),0),0)
  • {1,,1{1,,1{1,,1,,2}2◊2}1{1,,1{1,,1,,2}2◊2}1,2{1,,1,,2}2◊2} has recursion level C(C(Ω22+b+C(C(Ω22+C(Ω2+C(Ω22,0),a),0),a)+C(C(Ω22+b+C(C(Ω22+C(Ω2+C(Ω22,0),a),0),a)+1,0),C(Ω2+C(Ω22,0),0)),0),0)
  • {1,,1{1,,1{1,,1,,2}2◊2}1{1,,1{1,,1,,2}2◊2}1{1,,1{1,,1,,2}2}2{1,,1,,2}2◊2} has recursion level C(C(Ω22+b+C(C(Ω22+C(Ω2+C(Ω22,0),a),0),a)+C(C(Ω22+b+C(C(Ω22+C(Ω2+C(Ω22,0),a),0),a)+C(Ω2+C(Ω22,0),0),0),C(Ω2+C(Ω22,0),0)),0),0)
  • {1,,1{1,,1{1,,1,,2}2◊2}1{1,,1{1,,1,,2}2◊2}1{1,,2{1,,1,,2}2}2{1,,1,,2}2◊2} has recursion level C(C(Ω22+b+C(C(Ω22+C(Ω2+C(Ω22,0),a),0),a)+C(Ω2,C(Ω2+C(Ω22,0),0)),0),0)
  • {1,,1{1,,1{1,,1,,2}2◊2}1{1,,1{1,,1,,2}2◊2}1{1,,1{1,,1,,2}1,2}2{1,,1,,2}2◊2} has recursion level C(C(Ω22+b+C(C(Ω22+C(Ω2+C(Ω22,0),a),0),a)+C(Ω2+C(Ω22,0)+1,0),0),0)
  • {1,,1{1,,1{1,,1,,2}2◊2}1{1,,1{1,,1,,2}2◊2}1{1,,1◊2}2{1,,1,,2}2◊2} has recursion level C(C(Ω22+b+C(C(Ω22+C(Ω2+C(Ω22,0),a),0),a)+a,0),0)
  • {1,,1{1,,1{1,,1,,2}2◊2}1{1,,1{1,,1,,2}2◊2}1 {1{1,,1{1,,1{1,,1,,2}2◊2}1{1,,1{1,,1,,2}2◊2}2{1,,1,,2}2◊2}2,,1◊2}2{1,,1,,2}2◊2} has recursion level C(C(Ω22+b+C(C(Ω22+C(Ω2+C(Ω22,0),a),0),a)2,0),0)
  • {1,,1{1,,1{1,,1,,2}2◊2}1{1,,1{1,,1,,2}2◊2}1 {1{1,,1{1,,1{1,,1,,2}2◊2}1{1,,1{1,,1,,2}2◊2}2{1,,1,,2}2◊2}1,2,,1◊2}2{1,,1,,2}2◊2} has recursion level C(C(Ω22+b+C(C(Ω22+C(Ω2+C(Ω22,0),a),0)+1,a),0),0)
  • {1,,1{1,,1{1,,1,,2}2◊2}1{1,,1{1,,1,,2}2◊2}1 {1{1,,2{1,,1{1,,1,,2}2◊2}1{1,,1{1,,1,,2}2◊2}2{1,,1,,2}2◊2}2,,1◊2}2{1,,1,,2}2◊2} has recursion level C(C(Ω22+b+C(C(Ω2,C(Ω22+C(Ω2+C(Ω22,0),a),0)),a),0),0)
  • {1,,1{1,,1{1,,1,,2}2◊2}1{1,,1{1,,1,,2}2◊2}1 {1{1,,1{1,,1{1,,1,,2}2◊2}1{1,,1{1,,1,,2}2◊2}1,2{1,,1,,2}2◊2}2,,1◊2}2{1,,1,,2}2◊2} has recursion level C(C(Ω22+b+C(C(Ω22+C(Ω2+C(Ω22,0),a)+1,0),a),0),0)
  • {1,,1{1,,1{1,,1,,2}2◊2}1{1,,1{1,,1,,2}2◊2}1 {1{1,,1{1,,1{1,,1,,2}2◊2}1{1,,1{1,,1,,2}2◊2}1{1,,1◊2}2{1,,1,,2}2◊2}2,,1◊2}2{1,,1,,2}2◊2} has recursion level C(C(Ω22+b+C(C(Ω22+C(Ω2+C(Ω22,0),a)+a,0),a),0),0)
  • {1,,1{1,,1{1,,1,,2}2◊2}1{1,,1{1,,1,,2}2◊2}1{1,,2◊2}2{1,,1,,2}2◊2} has recursion level C(C(Ω22+b+C(C(Ω22+C(Ω2+C(Ω22,0),a)+C(Ω2,a),0),a),0),0)
  • {1,,1{1,,1{1,,1,,2}2◊2}1{1,,1{1,,1,,2}2◊2}1{1,,1{1,,1,,2}2◊2}2{1,,1,,2}2◊2} has recursion level C(C(Ω22+b+C(C(Ω22+C(Ω2+C(Ω22,0),a)2,0),a),0),0)
  • {1,,1{2,,1{1,,1,,2}2◊2}2{1,,1,,2}2◊2} has recursion level C(C(Ω22+b+C(C(Ω22+ω^(C(Ω2+C(Ω22,0),a)+1),0),a),0),0)
  • {1,,1{1{1,,2{1,,1,,2}2◊2}2,,1{1,,1,,2}2◊2}2{1,,1,,2}2◊2} has recursion level C(C(Ω22+b+C(C(Ω22+C(C(Ω2,C(Ω2+C(Ω22,0),a)),C(Ω2+C(Ω22,0),a)),0),a),0),0)
  • {1,,1{1{1,,1{1,,1{1,,1,,2}2◊2}2{1,,1,,2}2◊2}2,,1{1,,1,,2}2◊2}2{1,,1,,2}2◊2} has recursion level C(C(Ω22+b+C(C(Ω22+C(C(Ω2+C(Ω2+C(Ω22,0),a),C(Ω2+C(Ω22,0),a)),C(Ω2+C(Ω22,0),a)),0),a),0),0)
  • {1,,1{1,,1{1,,1,,2}2◊2}2{1,,1,,2}2◊2} has recursion level C(C(Ω22+b+C(C(Ω22+C(C(Ω2+C(Ω2,C(Ω2+C(Ω22,0),a)),C(Ω2+C(Ω22,0),a)),C(Ω2+C(Ω22,0),a)),0),a),0),0)
  • {1,,1{1,,1,,2}3◊2} has recursion level C(C(Ω22+b+C(C(Ω22+C(C(Ω22,0),C(Ω2+C(Ω22,0),a)),0),a),0),0)
  • {1,,1{1,,1{1,,1,,2}3◊2}1{1,,2{1,,1,,2}2◊2}2{1,,1,,2}3◊2} has recursion level C(C(Ω22+b+C(C(Ω22+C(Ω2,C(Ω2+C(Ω22,0),a)),0),a),0),0)
  • {1,,1{1,,1{1,,1,,2}3◊2}1{1,,1{1,,1,,2}3◊2}2{1,,1,,2}3◊2} has recursion level C(C(Ω22+b+C(C(Ω22+C(Ω2+C(Ω22,0),C(Ω2+C(Ω22,0),a)),0),a),0),0)
  • {1,,1{1,,1,,2}1,2◊2} has recursion level C(C(Ω22+b+C(C(Ω22+C(Ω2+C(Ω22,0)+1,a),0),a),0),0)
  • {1,,1{1,,1,,2}1{1,,1◊2}2◊2} has recursion level C(C(Ω22+b+C(C(Ω22+C(Ω2+C(Ω22,0)+a,a),0),a),0),0)
  • {1,,1{1,,1,,2}1{1,,1,,2}2◊2} has recursion level C(C(Ω22+b+C(C(Ω22+C(Ω2+C(Ω22,0)2,a),0),a),0),0)
  • {1,,1{1{1,,2,,2}2,,1,,2}2◊2} has recursion level C(C(Ω22+b+C(C(Ω22+C(Ω2+C(C(Ω2,C(Ω22,0)),C(Ω22,0)),a),0),a),0),0)
  • {1,,1{1{1,,1{1,,1,,2}2,,2}2,,1,,2}2◊2} has recursion level C(C(Ω22+b+C(C(Ω22+C(Ω2+C(C(Ω2+C(Ω22,0),C(Ω22,0)),C(Ω22,0)),a),0),a),0),0)
  • {1,,1{1{1,,1{1,,2,,2}2,,2}2,,1,,2}2◊2} has recursion level C(C(Ω22+b+C(C(Ω22+C(Ω2+C(C(Ω22,C(Ω22,0)),C(Ω22,0)),a),0),a),0),0)
  • {1,,1{1{1,,1{1,,2,,2}1,2,,2}2,,1,,2}2◊2} has recursion level C(C(Ω22+b+C(C(Ω22+C(Ω2+C(C(Ω22+1,0),C(Ω22,0)),a),0),a),0),0)
  • {1,,1{1{1,,1{1,,2,,2}1{1,,1{1{1,,1{1,,2,,2}1,2,,2}2,,1,,2}2}2,,2}2,,1,,2}2◊2} has recursion level C(C(Ω22+b+C(C(Ω22+C(Ω2+C(C(Ω22+C(Ω2+C(C(Ω22+1,0),C(Ω22,0)),0),0),C(Ω22,0)),a),0),a),0),0)
  • {1,,1{1{1,,1{1,,2,,2}1{1,,1◊2}2,,2}2,,1,,2}2◊2} has recursion level C(C(Ω22+b+C(C(Ω22+C(Ω2+C(C(Ω22+a,0),C(Ω22,0)),a),0),a),0),0)
  • {1,,1{1{1,,1{1,,2,,2}1{1,,1{1,,1,,2}2◊2}2,,2}2,,1,,2}2◊2} has recursion level C(C(Ω22+b+C(C(Ω22+C(Ω2+C(C(Ω22+C(Ω2+C(Ω22,0),a),0),C(Ω22,0)),a),0),a),0),0)
  • {1,,1{1{1,,1{1,,2,,2}1{1,,1{1{1,,1{1,,2,,2}1{1,,1◊2}2,,2}2,,1,,2}2◊2}2,,2}2,,1,,2}2◊2} has recursion level C(C(Ω22+b+C(C(Ω22+C(Ω2+C(C(Ω22+C(Ω2+C(C(Ω22+a,0),C(Ω22,0)),a),0),C(Ω22,0)),a),0),a),0),0)
  • {1,,1◊3} has recursion level C(C(Ω22+b+C(C(Ω22+b,0),a),0),0)
  • {1,,1{1,,1◊3}1{1,,2{1,,1,,2}2}2◊3} has recursion level C(C(Ω22+b+C(C(Ω22+b,0),a)+C(Ω2,C(Ω2+C(Ω22,0),0)),0),0)
  • {1,,1{1,,1◊3}1{1,,1◊2}2◊3} has recursion level C(C(Ω22+b+C(C(Ω22+b,0),a)+a,0),0)
  • {1,,1{1,,1◊3}1{1{1,,1◊3}2,,1◊2}2◊3} has recursion level C(C(Ω22+b+C(C(Ω22+b,0),a)2,0),0)
  • {1,,1{1,,1◊3}1{1{1,,1◊3}1,2,,1◊2}2◊3} has recursion level C(C(Ω22+b+C(C(Ω22+b,0)+1,a),0),0)
  • {1,,1{1,,1◊3}1{1{1{1,,1◊3}2,,1◊3}2,,1◊2}2◊3} has recursion level C(C(Ω22+b+C(C(Ω22+b,0)2,a),0),0)
  • {1,,1{1,,1◊3}1{1{1,,2◊3}2,,1◊2}2◊3} has recursion level C(C(Ω22+b+C(C(Ω2,C(Ω22+b,0)),a),0),0)
  • {1,,1{1,,1◊3}1{1{1,,1{1,,1◊2}2◊3}2,,1◊2}2◊3} has recursion level C(C(Ω22+b+C(C(Ω2+a,C(Ω22+b,0)),a),0),0)
  • {1,,1{1,,1◊3}1{1{1,,1{1,,1{1,,1,,2}2◊2}2◊3}2,,1◊2}2◊3} has recursion level C(C(Ω22+b+C(C(Ω22,C(Ω22+b,0)),a),0),0)
  • {1,,1{1,,1◊3}1{1{1,,1{1,,1◊3}2◊3}2,,1◊2}2◊3} has recursion level C(C(Ω22+b+C(C(Ω22+b,C(Ω22+b,0)),a),0),0)
  • {1,,1{1,,1◊3}1{1{1,,1{1,,1◊3}1,2◊3}2,,1◊2}2◊3} has recursion level C(C(Ω22+b+C(C(Ω22+b+1,0),a),0),0)
  • {1,,1{1,,1◊3}1{1{1,,1{1,,1◊3}1{1,,1◊2}2◊3}2,,1◊2}2◊3} has recursion level C(C(Ω22+b+C(C(Ω22+b+a,0),a),0),0)
  • {1,,1{1,,1◊3}1{1{1,,1{1,,1◊3}1{1{1,,1{1,,1◊3}1,2◊3}2,,1◊2}2◊3}2,,1◊2}2◊3} has recursion level C(C(Ω22+b+C(C(Ω22+b+C(C(Ω22+b+1,0),a),0),a),0),0)
  • {1,,1{1,,1◊3}1{1,,2◊2}2◊3} has recursion level C(C(Ω22+b+C(Ω2,a),0),0)
  • {1,,1{1,,1◊3}1{1,,1{1,,1,,2}2◊2}2◊3} has recursion level C(C(Ω22+b+C(Ω2+C(Ω22,0),a),0),0)
  • {1,,1{1,,1◊3}1{1,,1{1{1,,2,,2}2,,1,,2}2◊2}2◊3} has recursion level C(C(Ω22+b+C(Ω2+C(C(Ω2,C(Ω22,0)),C(Ω22,0)),a),0),0)
  • {1,,1{1,,1◊3}1{1,,1{1{1,,1{1,,2,,2}1,2,,2}2,,1,,2}2◊2}2◊3} has recursion level C(C(Ω22+b+C(Ω2+C(C(Ω22+1,0),C(Ω22,0)),a),0),0)
  • {1,,1{1,,1◊3}1{1,,1{1{1,,1{1,,2,,2}1{1,,1◊2}2,,2}2,,1,,2}2◊2}2◊3} has recursion level C(C(Ω22+b+C(Ω2+C(C(Ω22+a,0),C(Ω22,0)),a),0),0)
  • {1,,1{1,,1◊3}1{1,,1◊3}2◊3} has recursion level C(C(Ω22+b+C(Ω2+C(C(Ω22+b,0),C(Ω22,0)),a),0),0)
  • {1,,1◊4} has recursion level C(C(Ω22+b+C(C(Ω22+b,0),C(Ω2+C(C(Ω22+b,0),C(Ω22,0)),a)),0),0)
  • {1,,1{1,,1◊4}1{1,,1◊4}2◊4} has recursion level C(C(Ω22+b+C(Ω2+C(C(Ω22+b,0),C(Ω22,0)),C(Ω2+C(C(Ω22+b,0),C(Ω22,0)),a)),0),0)
  • {1,,1◊5} has recursion level C(C(Ω22+b+C(C(Ω22+b,0),C(Ω2+C(C(Ω22+b,0),C(Ω22,0)),C(Ω2+C(C(Ω22+b,0),C(Ω22,0)),a))),0),0)
  • {1,,1◊1,2} has recursion level C(C(Ω22+b+C(Ω2+C(C(Ω22+b,0),C(Ω22,0))+1,0),0),0)
  • {1,,1◊1{1,,1◊1,2}2} has recursion level C(C(Ω22+b+C(Ω2+C(C(Ω22+b,0),C(Ω22,0))+C(Ω2+C(C(Ω22+b,0),C(Ω22,0))+1,0),0),0),0)
  • {1,,1◊1{1,,1,,2}2} has recursion level C(C(Ω22+b+C(Ω2+C(C(Ω22+b,0),C(Ω22,0))+C(Ω22,0),0),0),0)
  • {1,,1◊1{1{1,,2,,2}2,,1,,2}2} has recursion level C(C(Ω22+b+C(Ω2+C(C(Ω22+b,0),C(Ω22,0))+C(C(Ω2,C(Ω22,0)),C(Ω22,0)),0),0),0)
  • {1,,1◊1{1{1,,1{1,,2,,2}2,,2}2,,1,,2}2} has recursion level C(C(Ω22+b+C(Ω2+C(C(Ω22+b,0),C(Ω22,0))+C(C(Ω22,C(Ω22,0)),C(Ω22,0)),0),0),0)
  • {1,,1◊1{1{1,,1{1,,2,,2}1,2,,2}2,,1,,2}2} has recursion level C(C(Ω22+b+C(Ω2+C(C(Ω22+b,0),C(Ω22,0))+C(C(Ω22+1,0),C(Ω22,0)),0),0),0)
  • {1,,1◊1{1{1,,1{1,,2,,2}1{1,,1◊1{1{1,,1{1,,2,,2}1,2,,2}2,,1,,2}2}2,,2}2,,1,,2}2} has recursion level C(C(Ω22+b+C(Ω2+C(C(Ω22+b,0),C(Ω22,0))+C(C(Ω22+C(Ω2+C(C(Ω22+b,0),C(Ω22,0))+C(C(Ω22+1,0),C(Ω22,0)),0),0),C(Ω22,0)),0),0),0)
  • {1,,1◊1◊2} has recursion level C(C(Ω22+b+C(Ω2+C(C(Ω22+b,0),C(Ω22,0))2,0),0),0)

Now let ordinal b = C(Ω2+C(Ω2,C(Ω22,0)),0).

  • {1,,1{2,,1{1,,2,,2}1{1,,1,,2}2,,2}2} has recursion level C(C(Ω22+b+C(Ω2+ω^(C(C(Ω22+b,0),C(Ω22,0))+1),0),0),0)
  • {1{1{1,,1{1,,2,,2}1{1,,1,,2}2,,2}2,,1,,2}2{1,,1{1,,2,,2}1{1,,1,,2}2,,2}2,,1,,2} has recursion level C(C(Ω22+b+C(Ω2+ω^(C(C(Ω22+b,0),C(Ω22,0))2),0),0),0)
  • {1{1,,2,,2}2{1,,1{1,,2,,2}1{1,,1,,2}2,,2}2,,1,,2} has recursion level C(C(Ω22+b+C(Ω2+C(C(Ω2,C(Ω22,0)),C(C(Ω22+b,0),C(Ω22,0))),0),0),0)
  • {1{1,,1{1,,2,,2}1{1,,1,,2}2,,2}3,,1,,2} has recursion level C(C(Ω22+b+C(Ω2+C(C(Ω22+b,0),C(C(Ω22+b,0),C(Ω22,0))),0),0),0)
  • {1{1,,1{1,,2,,2}1{1,,1,,2}2,,2}1,2,,1,,2} has recursion level C(C(Ω22+b+C(Ω2+C(C(Ω22+b,0)+1,C(Ω22,0)),0),0),0)
  • {1{1,,1{1,,2,,2}1{1,,1,,2}2,,2}1{1,,2,,2}2,,1,,2} has recursion level C(C(Ω22+b+C(Ω2+C(C(Ω22+b,0)+C(Ω2,C(Ω22,0)),C(Ω22,0)),0),0),0)
  • {1{1,,1,2,,2}2,,1{1,,2,,2}1{1,,1,,2}2,,2} has recursion level C(C(Ω22+b+C(Ω2+C(C(Ω22+b,0)+C(Ω2+1,C(Ω22,0)),C(Ω22,0)),0),0),0)
  • {1{1,,1{1,,2,,2}1,2,,2}2,,1{1,,2,,2}1{1,,1,,2}2,,2} has recursion level C(C(Ω22+b+C(Ω2+C(C(Ω22+b,0)+C(Ω22+1,0),C(Ω22,0)),0),0),0)
  • {1{1,,1{1,,2,,2}1{1,,1,,2}2,,2}2,,1{1,,2,,2}1{1,,1,,2}2,,2} has recursion level C(C(Ω22+b+C(Ω2+C(C(Ω22+b,0)2,C(Ω22,0)),0),0),0)
  • {1,,2{1,,2,,2}1{1,,1,,2}2,,2} has recursion level C(C(Ω22+b+C(Ω2+C(C(Ω2,C(Ω22+b,0)),C(Ω22,0)),0),0),0)
  • {1,,1{1,,1,,2}2{1,,2,,2}1{1,,1,,2}2,,2} has recursion level C(C(Ω22+b+C(Ω2+C(C(Ω2+C(Ω22,0),C(Ω22+b,0)),C(Ω22,0)),0),0),0)
  • {1,,1{1{1,,1,2{1,,2,,2}1{1,,1,,2}2,,2}2,,1,,2}2{1,,2,,2}1{1,,1,,2}2,,2} has recursion level C(C(Ω22+b+C(Ω2+C(C(Ω2+C(C(Ω2+1,C(Ω22+b,0)),C(Ω22,0)),C(Ω22+b,0)),C(Ω22,0)),0),0),0)
  • {1,,1{1,,2,,2}2{1,,1,,2}2,,2} has recursion level C(C(Ω22+b+C(Ω2+C(C(Ω22,C(Ω22+b,0)),C(Ω22,0)),0),0),0)
  • {1,,1{1,,2,,2}1,2{1,,1,,2}2,,2} has recursion level C(C(Ω22+b+C(Ω2+C(C(Ω22+1,C(Ω22+b,0)),C(Ω22,0)),0),0),0)
  • {1,,1{1,,2,,2}1{1,,1{1,,1{1,,2,,2}1,2{1,,1,,2}2,,2}2}2{1,,1,,2}2,,2} has recursion level C(C(Ω22+b+C(Ω2+C(C(Ω22+C(Ω2+C(C(Ω22+1,C(Ω22+b,0)),C(Ω22,0)),0),C(Ω22+b,0)),C(Ω22,0)),0),0),0)
  • {1,,1{1,,2,,2}1{1,,1,,2}3,,2} has recursion level C(C(Ω22+b+C(Ω2+C(C(Ω22+b,C(Ω22+b,0)),C(Ω22,0)),0),0),0)
  • {1,,1{1,,2,,2}1{1,,1,,2}1,2,,2} has recursion level C(C(Ω22+b+C(Ω2+C(C(Ω22+b+1,0),C(Ω22,0)),0),0),0)
  • {1,,1{1,,2,,2}1{1,,1,,2}1{1,,1{1,,1{1,,2,,2}1{1,,1,,2}1,2,,2}2}2,,2} has recursion level C(C(Ω22+b+C(Ω2+C(C(Ω22+b+C(Ω2+C(C(Ω22+b+1,0),C(Ω22,0)),0),0),C(Ω22,0)),0),0),0)
  • {1,,1{1,,2,,2}1{1,,1,,2}1{1,,1,,2}2,,2} has recursion level C(C(Ω22+b2,0),0)
  • {1,,1{1,,2,,2}1{2,,1,,2}2,,2} has recursion level C(C(Ω22+ω^(b+1),0),0)
  • {1,,1{1,,2,,2}1{1{1,,1{1,,1,,2}2}2,,1,,2}2,,2} has recursion level C(C(Ω22+ω^(b+C(Ω2+C(Ω22,0),0)),0),0)
  • {1,,1{1,,2,,2}1{1{1,,1{1,,1{1,,2,,2}1,2,,2}2}2,,1,,2}2,,2} has recursion level C(C(Ω22+ω^(b+C(Ω2+C(C(Ω22+1,0),C(Ω22,0)),0)),0),0)
  • {1,,1{1,,2,,2}1{1{1,,1{1,,1{1,,2,,2}1{2,,1,,2}2,,2}2}2,,1,,2}2,,2} has recursion level C(C(Ω22+ω^(b+C(Ω2+C(C(Ω22+ω^(b+1),0),C(Ω22,0)),0)),0),0)
  • {1,,1{1,,2,,2}1{1{1,,1,,2}2,,1,,2}2,,2} has recursion level C(C(Ω22+ω^(b2),0),0)
  • {1,,1{1,,2,,2}1{1{1,,2,,2}2,,1,,2}2,,2} has recursion level C(C(Ω22+C(C(Ω2,b),b),0),0)
  • {1,,1{1,,2,,2}1{1{1,,1{1,,1{1,,1,,2}2}2,,2}2,,1,,2}2,,2} has recursion level C(C(Ω22+C(C(Ω2+C(Ω2+C(Ω22,0),0),b),b),0),0)
  • {1,,1{1,,2,,2}1{1{1,,1{1,,1{1,,1{1,,2,,2}1{1,,1,,2}2,,2}2}2,,2}2,,1,,2}2,,2} has recursion level C(C(Ω22+C(C(Ω2+C(Ω2+C(C(Ω22+b,0),C(Ω22,0)),0),b),b),0),0)
  • {1,,1{1,,2,,2}1{1{1,,1{1,,1{1,,1{1,,2,,2}1{1{1,,2,,2}2,,1,,2}2,,2}2}2,,2}2,,1,,2}2,,2} has recursion level C(C(Ω22+C(C(Ω2+C(Ω2+C(C(Ω22+C(C(Ω2,b),b),0),C(Ω22,0)),0),b),b),0),0)
  • {1,,1{1,,2,,2}1{1{1,,1{1,,1,,2}2,,2}2,,1,,2}2,,2} has recursion level C(C(Ω22+C(C(Ω2+b,b),b),0),0)
  • {1,,1{1,,2,,2}1{1{1,,1{1,,2,,2}2,,2}2,,1,,2}2,,2} has recursion level C(C(Ω22+C(C(Ω2+C(Ω2,b),b),b),0),0)
  • {1,,1{1,,2,,2}1{1{1,,1{1,,2,,2}1,2,,2}2,,1,,2}2,,2} has recursion level C(C(Ω22+C(C(Ω2+C(Ω2,b)+1,b),b),0),0)
  • {1,,1{1,,2,,2}1{1,,2,,2}2,,2} has recursion level C(C(Ω22+C(C(Ω2+C(Ω2,b)2,b),b),0),0)
  • {1,,1{1,,1,2,,2}2,,2} has recursion level C(C(Ω22+C(C(Ω2+C(Ω2+1,b),b),b),0),0)
  • {1,,1{1,,1{1,,1,,2}2,,2}2,,2} has recursion level C(C(Ω22+C(C(Ω2+C(Ω2+b,b),b),b),0),0)
  • {1,,1,,3} has recursion level C(C(Ω22+C(C(Ω22,0),b),0),0)

So {1,,1,,3} has recursion level C(C(Ω22+C(C(Ω22,0),C(Ω2+C(Ω2,C(Ω22,0)),0)),0),0). Here’re some approximations, to make the comparisons above more simple:

  • {1,,1{1,,1{1,,1,,2}2,,2}2} approximately corresponds to C(Ω2+C(C(Ω2+C(Ω22,0),C(Ω22,0)),C(Ω22,0)),0)
  • {1,,1{1,,1{1,,2,,2}2,,2}2} approximately corresponds to C(Ω2+C(C(Ω22,C(Ω22,0)),C(Ω22,0)),0)
  • {1,,1{1,,1{1,,2,,2}1,2,,2}2} approximately corresponds to C(Ω2+C(C(Ω22+1,0),C(Ω22,0)),0)
  • {1,,1{1,,1{1,,2,,2}1{1,,1,,2}2,,2}2} approximately corresponds to C(Ω2+C(C(Ω22+C(Ω2+C(Ω2,C(Ω22,0)),0),0),C(Ω22,0)),0)
  • {1,,2{1,,1{1,,2,,2}1{1,,1,,2}2,,2}2} approximately corresponds to C(Ω2,C(Ω2+C(C(Ω22+C(Ω2+C(Ω2,C(Ω22,0)),0),0),C(Ω22,0)),0))
  • {1,,1{1,,1,,2}2{1,,1{1,,2,,2}1{1,,1,,2}2,,2}2} approximately corresponds to C(Ω2+C(Ω22,0),C(Ω2+C(C(Ω22+C(Ω2+C(Ω2,C(Ω22,0)),0),0),C(Ω22,0)),0))
  • {1,,1{1,,1{1,,2,,2}1{1,,1,,2}2,,2}1,2} approximately corresponds to C(Ω2+C(C(Ω22+C(Ω2+C(Ω2,C(Ω22,0)),0),0),C(Ω22,0))+1,0)
  • {1,,1{1,,1{1,,2,,2}1{1,,1,,2}1,2,,2}2} approximately corresponds to C(Ω2+C(C(Ω22+C(Ω2+C(Ω2,C(Ω22,0)),0)+1,0),C(Ω22,0)),0)
  • {1,,1{1,,1{1,,2,,2}1{1,,1,,2}1{1,,1,,2}2,,2}2} approximately corresponds to C(Ω2+C(C(Ω22+C(Ω2+C(Ω2,C(Ω22,0)),0)2,0),C(Ω22,0)),0)
  • {1,,1,,2} approximately corresponds to C(Ω2+C(Ω2,C(Ω22,0)),0)
  • {1,,2,,2} approximately corresponds to C(Ω2,C(Ω2+C(Ω2,C(Ω22,0)),0))
  • {1,,1{1,,1,,2}2,,2} approximately corresponds to C(Ω2+C(Ω2+C(Ω2,C(Ω22,0)),0),C(Ω2+C(Ω2,C(Ω22,0)),0))
  • {1,,1{1,,2,,2}2,,2} approximately corresponds to C(Ω2+C(Ω2,C(Ω2+C(Ω2,C(Ω22,0)),0)),C(Ω2+C(Ω2,C(Ω22,0)),0))

However, C(Ω22,0) still works in an irregular way.

Up to {1,,1,,1,,2}

Now let ordinal b = C(Ω2+C(Ω2,C(Ω22,0)),0).

  • {1,,1{1{1,,1{1,,1,,3}2}2,,1{1,,1,,2}2}2{1,,1,,2}2} has recursion level C(C(Ω22+C(C(Ω22,0),b),0),0)
  • {1,,1{1{1,,1{1,,1,,3}2}2,,1{1,,1,,2}2}1{1{1,,1{1,,1,,2}3}2,,1{1,,1,,2}2}2{1,,1,,2}2} has recursion level C(C(Ω22+C(C(Ω22,0),b)+C(C(Ω22,0),C(Ω2+C(Ω22,0),0)),0),0)
  • {1,,1{1{1,,1{1,,1,,3}2}2,,1{1,,1,,2}2}1{1{1,,1{1,,1,,2}1,2}2,,1{1,,1,,2}2}2{1,,1,,2}2} has recursion level C(C(Ω22+C(C(Ω22,0),b)+C(C(Ω22+C(Ω2+C(Ω22,0)+1,0),0),C(Ω2+C(Ω22,0),0)),0),0)
  • {1,,1{1{1,,1{1,,1,,3}2}2,,1{1,,1,,2}2}1{1{1,,1{1,,1{1,,2,,2}1,2,,2}2}2,,1{1,,1,,2}2}2{1,,1,,2}2} has recursion level C(C(Ω22+C(C(Ω22,0),b)+C(C(Ω22+C(Ω2+C(C(Ω22+1,0),C(Ω22,0)),0),0),C(Ω2+C(Ω22,0),0)),0),0)
  • {1,,1{1{1,,1{1,,1,,3}2}2,,1{1,,1,,2}2}1{1{1,,1{1,,1{1,,2,,2}1{1,,1,,2}2,,2}2}2,,1{1,,1,,2}2}2{1,,1,,2}2} has recursion level C(C(Ω22+C(C(Ω22,0),b)+C(C(Ω22+b,0),C(Ω2+C(Ω22,0),0)),0),0)
  • {1,,1{1{1,,1{1,,1,,3}2}2,,1{1,,1,,2}2}1 {1{1,,1{1,,1{1,,2,,2}1{1,,1,,2}1{1,,1,,2}2,,2}2}2,,1{1,,1,,2}2}2{1,,1,,2}2} has recursion level C(C(Ω22+C(C(Ω22,0),b)+C(C(Ω22+b2,0),C(Ω2+C(Ω22,0),0)),0),0)
  • {1,,1{1{1,,1{1,,1,,3}2}2,,1{1,,1,,2}2}1{1{1,,1{1,,1,,3}2}2,,1{1,,1,,2}2}2{1,,1,,2}2} has recursion level C(C(Ω22+C(C(Ω22,0),b)+C(C(Ω22+C(C(Ω22,0),b),0),C(Ω2+C(Ω22,0),0)),0),0)
  • {1,,1{1{1,,1{1,,1,,3}2}1,2,,1{1,,1,,2}2}2{1,,1,,2}2} has recursion level C(C(Ω22+C(C(Ω22,0),b)+C(C(Ω22+C(C(Ω22,0),b),0)+1,C(Ω2+C(Ω22,0),0)),0),0)
  • {1{1,,1{1,,1,,2}3}2,,1{1,,1,,3}2} has recursion level C(C(Ω22+C(C(Ω22,0),b)+C(C(Ω22+C(C(Ω22,0),b),0)+C(Ω22,0),C(Ω2+C(Ω22,0),0)),0),0)
  • {1{1,,1{1,,1{1,,2,,2}1{1,,1,,2}2,,2}2}2,,1{1,,1,,3}2} has recursion level C(C(Ω22+C(C(Ω22,0),b)+C(C(Ω22+C(C(Ω22,0),b),0)+C(Ω22+b,0),C(Ω2+C(Ω22,0),0)),0),0)
  • {1{1,,1{1,,1{1,,2,,2}1{1,,1,,2}1{1,,1,,2}2,,2}2}2,,1{1,,1,,3}2} has recursion level C(C(Ω22+C(C(Ω22,0),b)+C(C(Ω22+C(C(Ω22,0),b),0)+C(Ω22+b2,0),C(Ω2+C(Ω22,0),0)),0),0)
  • {1{1,,1{1,,1{1,,2,,2}1{1,,2,,2}2,,2}2}2,,1{1,,1,,3}2} has recursion level C(C(Ω22+C(C(Ω22,0),b)+C(C(Ω22+C(C(Ω22,0),b),0)+C(Ω22+C(C(Ω2+C(Ω2,b)2,b),b),0),C(Ω2+C(Ω22,0),0)),0),0)
  • {1{1,,1{1,,1,,3}2}2,,1{1,,1,,3}2} has recursion level C(C(Ω22+C(C(Ω22,0),b)+C(C(Ω22+C(C(Ω22,0),b),0)2,C(Ω2+C(Ω22,0),0)),0),0)
  • {1,,2{1,,1,,3}2} has recursion level C(C(Ω22+C(C(Ω22,0),b)+C(C(Ω2,C(Ω22+C(C(Ω22,0),b),0)),C(Ω2+C(Ω22,0),0)),0),0)
  • {1,,1{1,,1{1,,1,,2}3}2{1,,1,,3}2} has recursion level C(C(Ω22+C(C(Ω22,0),b)+C(C(Ω22,C(Ω22+C(C(Ω22,0),b),0)),C(Ω2+C(Ω22,0),0)),0),0)
  • {1,,1{1,,1{1,,1{1,,2,,2}1{1,,1,,2}2,,2}2}2{1,,1,,3}2} has recursion level C(C(Ω22+C(C(Ω22,0),b)+C(C(Ω22+b,C(Ω22+C(C(Ω22,0),b),0)),C(Ω2+C(Ω22,0),0)),0),0)
  • {1,,1{1,,1{1,,1,,3}2}2{1,,1,,3}2} has recursion level C(C(Ω22+C(C(Ω22,0),b)+C(C(Ω22+C(C(Ω22,0),b),C(Ω22+C(C(Ω22,0),b),0)),C(Ω2+C(Ω22,0),0)),0),0)
  • {1,,1{1,,1{1,,1,,3}2}1,2{1,,1,,3}2} has recursion level C(C(Ω22+C(C(Ω22,0),b)+C(C(Ω22+C(C(Ω22,0),b)+1,0),C(Ω2+C(Ω22,0),0)),0),0)
  • {1,,1{1,,1{1,,1,,3}2}1{1,,1{1,,1,,2}2}2{1,,1,,3}2} has recursion level C(C(Ω22+C(C(Ω22,0),b)+C(C(Ω22+C(C(Ω22,0),b)+C(Ω2+C(Ω22,0),0),0),C(Ω2+C(Ω22,0),0)),0),0)
  • {1,,1{1,,1{1,,1,,3}2}1{1,,2{1,,1,,2}2}2{1,,1,,3}2} has recursion level C(C(Ω22+C(C(Ω22,0),b)+C(Ω2,C(Ω2+C(Ω22,0),0)),0),0)
  • {1,,1{1,,1{1,,1,,3}2}1{1,,1{1,,1{1,,2,,2}1,2,,2}2}2{1,,1,,3}2} has recursion level C(C(Ω22+C(C(Ω22,0),b)+C(Ω2+C(C(Ω22+1,0),C(Ω22,0)),0),0),0)
  • {1,,1{1,,1{1,,1,,3}2}1{1,,1{1,,1{1,,2,,2}1{1,,1,,2}2,,2}2}2{1,,1,,3}2} has recursion level C(C(Ω22+C(C(Ω22,0),b)+C(Ω2+C(C(Ω22+b,0),C(Ω22,0)),0),0),0)
  • {1,,1{1,,1{1,,1,,3}2}1{1,,1{1,,1{1,,2,,2}1{1,,1,,2}1{1,,1,,2}2,,2}2}2{1,,1,,3}2} has recursion level C(C(Ω22+C(C(Ω22,0),b)+C(Ω2+C(C(Ω22+b2,0),C(Ω22,0)),0),0),0)
  • {1,,1{1,,1{1,,1,,3}2}1{1,,1{1,,1,,3}2}2{1,,1,,3}2} has recursion level C(C(Ω22+C(C(Ω22,0),b)+C(Ω2+C(C(Ω22+C(C(Ω22,0),b),0),C(Ω22,0)),0),0),0)
  • {1,,1{1,,1,,2}2{1,,1,,3}2} has recursion level C(C(Ω22+C(C(Ω22,0),b)+C(C(Ω22,0),C(Ω2+C(C(Ω22+C(C(Ω22,0),b),0),C(Ω22,0)),0)),0),0)
  • {1,,1{1,,1,,2}1,2{1,,1,,3}2} has recursion level C(C(Ω22+C(C(Ω22,0),b)+C(Ω2+C(Ω22,0)+1,C(Ω2+C(C(Ω22+C(C(Ω22,0),b),0),C(Ω22,0)),0)),0),0)
  • {1,,1{1,,2,,2}2{1,,1,,3}2} has recursion level C(C(Ω22+C(C(Ω22,0),b)+C(Ω2+C(C(Ω2,C(Ω22,0)),C(Ω22,0)),C(Ω2+C(C(Ω22+C(C(Ω22,0),b),0),C(Ω22,0)),0)),0),0)
  • {1,,1{1,,1,,3}3} has recursion level C(C(Ω22+C(C(Ω22,0),b)+C(Ω2+C(C(Ω22+C(C(Ω22,0),b),0),C(Ω22,0)),C(Ω2+C(C(Ω22+C(C(Ω22,0),b),0),C(Ω22,0)),0)),0),0)
  • {1,,1{1,,1,,3}1,2} has recursion level C(C(Ω22+C(C(Ω22,0),b)+C(Ω2+C(C(Ω22+C(C(Ω22,0),b),0),C(Ω22,0))+1,0),0),0)
  • {1,,1{1,,1,,3}1{1,,1,,3}2} has recursion level C(C(Ω22+C(C(Ω22,0),b)+C(Ω2+C(C(Ω22+C(C(Ω22,0),b),0),C(Ω22,0))2,0),0),0)
  • {1,,1{2,,1{1,,1,,3}2,,2}2} has recursion level C(C(Ω22+C(C(Ω22,0),b)+C(Ω2+ω^(C(C(Ω22+C(C(Ω22,0),b),0),C(Ω22,0))+1),0),0),0)
  • {1{1,,2,,2}2{1,,1{1,,1,,3}2,,2}2,,1,,2} has recursion level C(C(Ω22+C(C(Ω22,0),b)+C(Ω2+C(C(Ω2,C(Ω22,0)),C(C(Ω22+C(C(Ω22,0),b),0),C(Ω22,0))),0),0),0)
  • {1{1,,1{1,,2,,2}1,2,,2}2{1,,1{1,,1,,3}2,,2}2,,1,,2} has recursion level C(C(Ω22+C(C(Ω22,0),b)+C(Ω2+C(C(Ω22+1,0),C(C(Ω22+C(C(Ω22,0),b),0),C(Ω22,0))),0),0),0)
  • {1{1,,1{1,,2,,2}1{1,,1,,2}2,,2}2{1,,1{1,,1,,3}2,,2}2,,1,,2} has recursion level C(C(Ω22+C(C(Ω22,0),b)+C(Ω2+C(C(Ω22+b,0),C(C(Ω22+C(C(Ω22,0),b),0),C(Ω22,0))),0),0),0)
  • {1{1,,1{1,,1,,3}2,,2}3,,1,,2} has recursion level C(C(Ω22+C(C(Ω22,0),b)+C(Ω2+C(C(Ω22+C(C(Ω22,0),b),0),C(C(Ω22+C(C(Ω22,0),b),0),C(Ω22,0))),0),0),0)
  • {1{1,,1{1,,1,,3}2,,2}1,2,,1,,2} has recursion level C(C(Ω22+C(C(Ω22,0),b)+C(Ω2+C(C(Ω22+C(C(Ω22,0),b),0)+1,C(Ω22,0)),0),0),0)
  • {1{1,,1{1,,1,,3}2,,2}1{1,,1,,2}2,,1,,2} has recursion level C(C(Ω22+C(C(Ω22,0),b)+C(Ω2+C(C(Ω22+C(C(Ω22,0),b),0)+C(Ω22,0),C(Ω22,0)),0),0),0)
  • {1{1,,1{1,,2,,2}1{1,,1,,2}2,,2}2,,1{1,,2,,2}1{1{1,,1{1,,1,,3}2,,2}2,,1,,2}2,,2} has recursion level C(C(Ω22+C(C(Ω22,0),b)+C(Ω2+C(C(Ω22+C(C(Ω22,0),b),0)+C(Ω22+b,0),C(Ω22,0)),0),0),0)
  • {1{1,,1{1,,2,,2}1{1{1,,1{1,,1,,3}2,,2}2,,1,,2}2,,2}2,,1{1,,2,,2}1{1{1,,1{1,,1,,3}2,,2}2,,1,,2}2,,2} has recursion level C(C(Ω22+C(C(Ω22,0),b)+C(Ω2+C(C(Ω22+C(C(Ω22,0),b),0)2,C(Ω22,0)),0),0),0)
  • {1,,2{1,,2,,2}1{1{1,,1{1,,1,,3}2,,2}2,,1,,2}2,,2} has recursion level C(C(Ω22+C(C(Ω22,0),b)+C(Ω2+C(C(Ω2,C(Ω22+C(C(Ω22,0),b),0)),C(Ω22,0)),0),0),0)
  • {1,,1{1,,2,,2}2{1{1,,1{1,,1,,3}2,,2}2,,1,,2}2,,2} has recursion level C(C(Ω22+C(C(Ω22,0),b)+C(Ω2+C(C(Ω22,C(Ω22+C(C(Ω22,0),b),0)),C(Ω22,0)),0),0),0)
  • {1,,1{1,,2,,2}1{1,,1,,2}2{1{1,,1{1,,1,,3}2,,2}2,,1,,2}2,,2} has recursion level C(C(Ω22+C(C(Ω22,0),b)+C(Ω2+C(C(Ω22+b,C(Ω22+C(C(Ω22,0),b),0)),C(Ω22,0)),0),0),0)
  • {1,,1{1,,2,,2}1{1{1,,1{1,,1,,3}2,,2}2,,1,,2}3,,2} has recursion level C(C(Ω22+C(C(Ω22,0),b)+C(Ω2+C(C(Ω22+C(C(Ω22,0),b),C(Ω22+C(C(Ω22,0),b),0)),C(Ω22,0)),0),0),0)
  • {1,,1{1,,2,,2}1{1{1,,1{1,,1,,3}2,,2}2,,1,,2}1,2,,2} has recursion level C(C(Ω22+C(C(Ω22,0),b)+C(Ω2+C(C(Ω22+C(C(Ω22,0),b)+1,0),C(Ω22,0)),0),0),0)
  • {1,,1{1,,2,,2}1{1{1,,1{1,,1,,3}2,,2}2,,1,,2}1{1,,1,,2}2,,2} has recursion level C(C(Ω22+C(C(Ω22,0),b)+b,0),0)
  • {1,,1{1,,2,,2}1{1{1,,1{1,,1,,3}2,,2}2,,1,,2}1{1{1,,2,,2}2,,1,,2}2,,2} has recursion level C(C(Ω22+C(C(Ω22,0),b)+C(C(Ω2,b),b),0),0)
  • {1,,1{1,,2,,2}1{1{1,,1{1,,1,,3}2,,2}2,,1,,2}1{1{1,,1{1,,1,,3}2,,2}2,,1,,2}2,,2} has recursion level C(C(Ω22+C(C(Ω22,0),b)2,0),0)
  • {1,,1{1,,2,,2}1{1{1,,2,,2}2{1,,1{1,,1,,3}2,,2}2,,1,,2}2,,2} has recursion level C(C(Ω22+C(C(Ω2,b),C(C(Ω22,0),b)),0),0)
  • {1,,1{1,,2,,2}1{1{1,,1{1,,1,,3}2,,2}3,,1,,2}2,,2} has recursion level C(C(Ω22+C(C(Ω22,0),C(C(Ω22,0),b)),0),0)
  • {1,,1{1,,2,,2}1{1{1,,1{1,,1,,3}2,,2}1,2,,1,,2}2,,2} has recursion level C(C(Ω22+C(C(Ω22,0)+1,b),0),0)
  • {1,,1{1,,2,,2}1{1{1,,1{1,,1,,3}2,,2}1{1,,2,,2}2,,1,,2}2,,2} has recursion level C(C(Ω22+C(C(Ω22,0)+C(Ω2,b),b),0),0)
  • {1,,1{1,,2,,2}1{1{1{1,,1,2,,2}2,,1{1,,1,,3}2,,2}2,,1,,2}2,,2} has recursion level C(C(Ω22+C(C(Ω22,0)+C(Ω2+1,b),b),0),0)
  • {1{1,,1{1,,1,,3}2,,2}2,,1{1,,1,,3}2,,2} has recursion level C(C(Ω22+C(C(Ω22,0)2,b),0),0)
  • {1,,2{1,,1,,3}2,,2} has recursion level C(C(Ω22+C(C(Ω2,C(Ω22,0)),b),0),0)
  • {1,,1,2{1,,1,,3}2,,2} has recursion level C(C(Ω22+C(C(Ω2+1,C(Ω22,0)),b),0),0)
  • {1,,1{1,,1,,2}2{1,,1,,3}2,,2} has recursion level C(C(Ω22+C(C(Ω2+b,C(Ω22,0)),b),0),0)
  • {1,,1{1,,2,,2}2{1,,1,,3}2,,2} has recursion level C(C(Ω22+C(C(Ω2+C(Ω2,b),C(Ω22,0)),b),0),0)
  • {1,,1{1,,1{1,,1,,2}2,,2}2{1,,1,,3}2,,2} has recursion level C(C(Ω22+C(C(Ω2+C(Ω2+b,b),C(Ω22,0)),b),0),0)
  • {1,,1{1,,1{1,,1,,3}2,,2}2{1,,1,,3}2,,2} has recursion level C(C(Ω22+C(C(Ω22,C(Ω22,0)),b),0),0)
  • {1,,1{1,,1{1,,1,,3}2,,2}1,2{1,,1,,3}2,,2} has recursion level C(C(Ω22+C(C(Ω22+1,0),b),0),0)
  • {1,,1{1,,1{1,,1,,3}2,,2}1{1,,1,,2}2{1,,1,,3}2,,2} has recursion level C(C(Ω22+C(C(Ω22+b,0),b),0),0)
  • {1,,1{1,,1{1,,1,,3}2,,2}1{1{1,,1{1,,1{1,,1,,3}2,,2}1,2{1,,1,,3}2,,2}2,,1,,2}2{1,,1,,3}2,,2} has recursion level C(C(Ω22+C(C(Ω22+C(C(Ω22+1,0),b),0),b),0),0)
  • {1,,1{1,,1{1,,1,,3}2,,2}1{1,,2,,2}2{1,,1,,3}2,,2} has recursion level C(C(Ω22+C(Ω2,b),0),0)
  • {1,,1{1,,1{1,,1,,3}2,,2}1{1,,1{1,,2,,2}2,,2}2{1,,1,,3}2,,2} has recursion level C(C(Ω22+C(Ω2+C(Ω2,b),b),0),0)
  • {1,,1{1,,1{1,,1,,3}2,,2}1{1,,1{1,,1,,3}2,,2}2{1,,1,,3}2,,2} has recursion level C(C(Ω22+C(Ω2+C(Ω22,0),b),0),0)
  • {1,,1{1{1,,2{1,,1,,3}2,,2}2,,1{1,,1,,3}2,,2}2{1,,1,,3}2,,2} has recursion level C(C(Ω22+C(C(Ω2,C(Ω2+C(Ω22,0),b)),C(Ω2+C(Ω22,0),b)),0),0)
  • {1,,1{1,,2{1,,1,,3}2,,2}2{1,,1,,3}2,,2} has recursion level C(C(Ω22+C(C(Ω2+C(Ω2,C(Ω2+C(Ω22,0),b)),C(Ω2+C(Ω22,0),b)),C(Ω2+C(Ω22,0),b)),0),0)
  • {1,,1{1,,1,,3}3,,2} has recursion level C(C(Ω22+C(C(Ω22,0),C(Ω2+C(Ω22,0),b)),0),0)
  • {1,,1{1,,1{1,,1,,3}3,,2}1{1,,1{1,,1,,3}3,,2}2{1,,1,,3}3,,2} has recursion level C(C(Ω22+C(Ω2+C(Ω22,0),C(Ω2+C(Ω22,0),b)),0),0)
  • {1,,1{1,,1,,3}1,2,,2} has recursion level C(C(Ω22+C(Ω2+C(Ω22,0)+1,b),0),0)
  • {1,,1{1,,1,,3}1{1,,1{1,,1{1,,1,,3}1,2,,2}2}2,,2} has recursion level C(C(Ω22+C(Ω2+C(Ω22,0)+C(Ω2+C(C(Ω22+C(Ω2+C(Ω22,0)+1,b),0),C(Ω22,0)),0),b),0),0)
  • {1,,1{1,,1,,3}1{1,,1,,2}2,,2} has recursion level C(C(Ω22+C(Ω2+C(Ω22,0)+b,b),0),0)
  • {1,,1{1,,1,,3}1{1,,2,,2}2,,2} has recursion level C(C(Ω22+C(Ω2+C(Ω22,0)+C(Ω2,b),b),0),0)
  • {1,,1{1,,1,,3}1{1,,1{1,,1,,3}2,,2}2,,2} has recursion level C(C(Ω22+C(Ω2+C(Ω22,0)+C(Ω2+C(Ω22,0),b),b),0),0)
  • {1,,1{1,,1,,3}1{1,,1{1,,1,,3}1,2,,2}2,,2} has recursion level C(C(Ω22+C(Ω2+C(Ω22,0)+C(Ω2+C(Ω22,0)+1,b),b),0),0)
  • {1,,1{1,,1,,3}1{1,,1,,3}2,,2} has recursion level C(C(Ω22+C(Ω2+C(Ω22,0)2,b),0),0)
  • {1,,1{2,,1,,3}2,,2} has recursion level C(C(Ω22+C(Ω2+ω^(C(Ω22,0)+1),b),0),0)
  • {1{1,,1,,3}2,,1,,3} has recursion level C(C(Ω22+C(Ω2+ω^(C(Ω22,0)2),b),0),0)
  • {1,,2,,3} has recursion level C(C(Ω22+C(Ω2+C(C(Ω2,C(Ω22,0)),C(Ω22,0)),b),0),0)
  • {1,,1,2,,3} has recursion level C(C(Ω22+C(Ω2+C(C(Ω2+1,C(Ω22,0)),C(Ω22,0)),b),0),0)
  • {1,,1{1,,1,,2}2,,3} has recursion level C(C(Ω22+C(Ω2+C(C(Ω2+b,C(Ω22,0)),C(Ω22,0)),b),0),0)
  • {1,,1{1,,1,,3}2,,3} has recursion level C(C(Ω22+C(Ω2+C(C(Ω2+C(Ω22,0),C(Ω22,0)),C(Ω22,0)),b),0),0)
  • {1,,1{1,,2,,3}2,,3} has recursion level C(C(Ω22+C(Ω2+C(C(Ω22,C(Ω22,0)),C(Ω22,0)),b),0),0)
  • {1,,1{1,,2,,3}1,2,,3} has recursion level C(C(Ω22+C(Ω2+C(C(Ω22+1,0),C(Ω22,0)),b),0),0)
  • {1,,1{1,,2,,3}1{1,,1,,2}2,,3} has recursion level C(C(Ω22+C(Ω2+C(C(Ω22+b,0),C(Ω22,0)),b),0),0)
  • {1,,1{1,,2,,3}1{1,,1{1,,1{1,,2,,3}1{1,,1,,2}2,,3}2,,2}2,,3} has recursion level C(C(Ω22+C(Ω2+C(C(Ω22+C(Ω2+C(C(Ω22+b,0),C(Ω22,0)),b),0),C(Ω22,0)),b),0),0)
  • {1,,1{1,,2,,3}1{1,,1,,3}2,,3} has recursion level C(C(Ω22+C(Ω2+C(Ω2,C(Ω22,0)),b),0),0)

Now let separator ♦ = {1,,1{1,,2,,3}1{1,,1,,3}2,,3} and let ordinal a = C(Ω2+C(Ω2,C(Ω22,0)),0), and b = C(Ω2+C(Ω2,C(Ω22,0)),a) = C(Ω2+C(Ω2,C(Ω22,0)),C(Ω2+C(Ω2,C(Ω22,0)),0)).

  • {1,,1{1♦2,,1{1,,1,,2}2}2{1,,1,,2}2} has recursion level C(C(Ω22+b,0),0)
  • {1,,1{2♦2,,1{1,,1,,2}2}2{1,,1,,2}2} has recursion level C(C(Ω22+b+ω^(C(C(Ω22+b,0),C(Ω2+C(Ω22,0),0))+1),0),0)
  • {1,,1{1♦1,2,,1{1,,1,,2}2}2{1,,1,,2}2} has recursion level C(C(Ω22+b+C(C(Ω22+b,0)+1,C(Ω2+C(Ω22,0),0)),0),0)
  • {1{1,,1{1,,1{1,,2,,2}1{1,,1,,2}2,,2}2}2,,1♦2} has recursion level C(C(Ω22+b+C(C(Ω22+b,0)+C(Ω22+a,0),C(Ω2+C(Ω22,0),0)),0),0)
  • {1{1,,1{1,,1,,3}2}2,,1♦2} has recursion level C(C(Ω22+b+C(C(Ω22+b,0)+C(Ω22+C(C(Ω22,0),a),0),C(Ω2+C(Ω22,0),0)),0),0)
  • {1{1,,1{1,,1{1,,1{1,,1,,3}2,,2}1,2{1,,1,,3}2,,2}2}2,,1♦2} has recursion level C(C(Ω22+b+C(C(Ω22+b,0)+C(Ω22+C(C(Ω22+1,0),a),0),C(Ω2+C(Ω22,0),0)),0),0)
  • {1{1,,1{1,,1{1,,1{1,,1,,3}2,,2}1{1,,2,,2}2{1,,1,,3}2,,2}2}2,,1♦2} has recursion level C(C(Ω22+b+C(C(Ω22+b,0)+C(Ω22+C(Ω2,a),0),C(Ω2+C(Ω22,0),0)),0),0)
  • {1{1,,1{1,,1{1,,1,,3}1,2,,2}2}2,,1♦2} has recursion level C(C(Ω22+b+C(C(Ω22+b,0)+C(Ω22+C(Ω2+C(Ω22,0)+1,a),0),C(Ω2+C(Ω22,0),0)),0),0)
  • {1{1,,1♦2}2,,1♦2} has recursion level C(C(Ω22+b+C(C(Ω22+b,0)2,C(Ω2+C(Ω22,0),0)),0),0)
  • {1,,2♦2} has recursion level C(C(Ω22+b+C(C(Ω2,C(Ω22+b,0)),C(Ω2+C(Ω22,0),0)),0),0)
  • {1,,1{1,,1{1,,1,,2}3}2♦2} has recursion level C(C(Ω22+b+C(C(Ω22,C(Ω22+b,0)),C(Ω2+C(Ω22,0),0)),0),0)
  • {1,,1{1,,1{1,,1{1,,2,,2}1{1,,1,,2}2,,2}2}2♦2} has recursion level C(C(Ω22+b+C(C(Ω22+a,C(Ω22+b,0)),C(Ω2+C(Ω22,0),0)),0),0)
  • {1,,1{1,,1♦2}2♦2} has recursion level C(C(Ω22+b+C(C(Ω22+b,C(Ω22+b,0)),C(Ω2+C(Ω22,0),0)),0),0)
  • {1,,1{1,,1♦2}1,2♦2} has recursion level C(C(Ω22+b+C(C(Ω22+b+1,0),C(Ω2+C(Ω22,0),0)),0),0)
  • {1,,1{1,,1♦2}1{1,,2{1,,1,,2}2}2♦2} has recursion level C(C(Ω22+b+C(Ω2,C(Ω2+C(Ω22,0),0)),0),0)
  • {1,,1{1,,1♦2}1{1,,1{1,,1,,2}1,2}2♦2} has recursion level C(C(Ω22+b+C(Ω2+C(Ω22,0)+1,0),0),0)
  • {1,,1{1,,1♦2}1{1,,1{1,,1{1,,2,,2}1{1,,1,,2}2,,2}2}2♦2} has recursion level C(C(Ω22+b+C(Ω2+C(C(Ω22+a,0),C(Ω22,0)),0),0),0)
  • {1,,1{1,,1♦2}1{1,,1{1,,1,,3}2}2♦2} has recursion level C(C(Ω22+b+C(Ω2+C(C(Ω22+C(C(Ω22,0),a),0),C(Ω22,0)),0),0),0)
  • {1,,1{1,,1♦2}1{1,,1{1,,1{1,,1{1,,1,,3}2,,2}1{1,,2,,2}2{1,,1,,3}2,,2}2}2 ♦2} has recursion level C(C(Ω22+b+C(Ω2+C(C(Ω22+C(Ω2,a),0),C(Ω22,0)),0),0),0)
  • {1,,1{1,,1♦2}1{1,,1{1,,1{1,,1{1,,1,,3}2,,2}1{1,,1{1,,1,,3}2,,2}2{1,,1,,3}2,,2}2}2♦2} has recursion level C(C(Ω22+b+C(Ω2+C(C(Ω22+C(Ω2+C(Ω22,0),a),0),C(Ω22,0)),0),0),0)
  • {1,,1{1,,1♦2}1{1,,1♦2}2♦2} has recursion level C(C(Ω22+b+C(Ω2+C(C(Ω22+b,0),C(Ω22,0)),0),0),0)
  • {1,,1{1,,1,,2}2♦2} has recursion level C(C(Ω22+b+C(C(Ω22,0),C(Ω2+C(C(Ω22+b,0),C(Ω22,0)),0)),0),0)
  • {1,,1{1,,1,,2}1,2♦2} has recursion level C(C(Ω22+b+C(C(Ω22+C(Ω2+C(Ω22,0)+1,C(Ω2+C(C(Ω22+b,0),C(Ω22,0)),0)),0),C(Ω2+C(C(Ω22+b,0),C(Ω22,0)),0)),0),0)
  • {1,,1{1,,1{1,,2,,2}1,2,,2}2♦2} has recursion level C(C(Ω22+b+C(C(Ω22+C(Ω2+C(C(Ω22+1,0),C(Ω22,0)),C(Ω2+C(C(Ω22+b,0),C(Ω22,0)),0)),0),C(Ω2+C(C(Ω22+b,0),C(Ω22,0)),0)),0),0)
  • {1,,1{1,,1{1,,2,,2}1{1,,1,,2}2,,2}2♦2} has recursion level C(C(Ω22+b+C(C(Ω22+a,0),C(Ω2+C(C(Ω22+b,0),C(Ω22,0)),0)),0),0)
  • {1,,1{1,,1{1,,1{1,,2,,2}1{1,,1,,2}2,,2}2♦2}1{1,,1♦2}2{1,,1{1,,2,,2}1{1,,1,,2}2,,2}2♦2} has recursion level C(C(Ω22+b+C(C(Ω22+a,0),C(Ω2+C(C(Ω22+b,0),C(Ω22,0)),0))+C(Ω2+C(C(Ω22+b,0),C(Ω22,0)),0),0),0)
  • {1,,1{1,,1{1,,1{1,,2,,2}1{1,,1,,2}2,,2}2♦2}1{1{1,,1{1,,1{1,,2,,2}1{1,,1,,2}2,,2}2♦2}2,,1♦2} 2{1,,1{1,,2,,2}1{1,,1,,2}2,,2}2♦2} has recursion level C(C(Ω22+b+C(C(Ω22+a,0),C(Ω2+C(C(Ω22+b,0),C(Ω22,0)),0))2,0),0)
  • {1,,1{1,,1{1,,1{1,,2,,2}1{1,,1,,2}2,,2}2♦2}1{1{1,,1{1,,1{1,,2,,2}1{1,,1,,2}2,,2}2♦2}1,2,,1♦2} 2{1,,1{1,,2,,2}1{1,,1,,2}2,,2}2♦2} has recursion level C(C(Ω22+b+C(C(Ω22+a,0)+1,C(Ω2+C(C(Ω22+b,0),C(Ω22,0)),0)),0),0)
  • {1,,1{1,,1{1,,1{1,,2,,2}1{1,,1,,2}2,,2}2♦2}1 {1{1,,1{1,,1{1,,1{1,,2,,2}1{1,,1,,2}2,,2}2♦2}1,2{1,,1{1,,2,,2}1{1,,1,,2}2,,2}2♦2}2,,1♦2} 2{1,,1{1,,2,,2}1{1,,1,,2}2,,2}2♦2} has recursion level C(C(Ω22+b+C(C(Ω22+a+1,0),C(Ω2+C(C(Ω22+b,0),C(Ω22,0)),0)),0),0)
  • {1,,1{1,,1{1,,1{1,,2,,2}1{1,,1,,2}2,,2}2♦2}1{1,,2♦2}2{1,,1{1,,2,,2}1{1,,1,,2}2,,2}2♦2} has recursion level C(C(Ω22+b+C(C(Ω22+a+C(Ω2,C(Ω2+C(C(Ω22+b,0),C(Ω22,0)),0)),0),C(Ω2+C(C(Ω22+b,0),C(Ω22,0)),0)),0),0)
  • {1,,1{1,,1{1,,2,,2}1{1,,1,,2}2,,2}3♦2} has recursion level C(C(Ω22+b+C(C(Ω22+a2,0),C(Ω2+C(C(Ω22+b,0),C(Ω22,0)),0)),0),0)
  • {1,,1{1,,1,,3}2♦2} has recursion level C(C(Ω22+b+C(C(Ω22+C(C(Ω22,0),a),0),C(Ω2+C(C(Ω22+b,0),C(Ω22,0)),0)),0),0)
  • {1,,1♦3} has recursion level C(C(Ω22+b+C(C(Ω22+b,0),C(Ω2+C(C(Ω22+b,0),C(Ω22,0)),0)),0),0)
  • {1,,1{1,,1♦3}1{1,,1♦2}2♦3} has recursion level C(C(Ω22+b+C(C(Ω22+b,0),C(Ω2+C(C(Ω22+b,0),C(Ω22,0)),0))+C(Ω2+C(C(Ω22+b,0),C(Ω22,0)),0),0),0)
  • {1,,1{1,,1♦3}1{1{1,,1♦3}2,,1♦2}2♦3} has recursion level C(C(Ω22+b+C(C(Ω22+b,0),C(Ω2+C(C(Ω22+b,0),C(Ω22,0)),0))2,0),0)
  • {1,,1{1,,1♦3}1{1{1,,1♦3}1,2,,1♦2}2♦3} has recursion level C(C(Ω22+b+C(C(Ω22+b,0)+1,C(Ω2+C(C(Ω22+b,0),C(Ω22,0)),0)),0),0)
  • {1,,1{1,,1♦3}1{1{1,,2♦3}2,,1♦2}2♦3} has recursion level C(C(Ω22+b+C(C(Ω2,C(Ω22+b,0)),C(Ω2+C(C(Ω22+b,0),C(Ω22,0)),0)),0),0)
  • {1,,1{1,,1♦3}1{1{1,,1{1,,1♦3}2♦3}2,,1♦2}2♦3} has recursion level C(C(Ω22+b+C(C(Ω22+b,C(Ω22+b,0)),C(Ω2+C(C(Ω22+b,0),C(Ω22,0)),0)),0),0)
  • {1,,1{1,,1♦3}1{1{1,,1{1,,1♦3}1,2♦3}2,,1♦2}2♦3} has recursion level C(C(Ω22+b+C(C(Ω22+b+1,0),C(Ω2+C(C(Ω22+b,0),C(Ω22,0)),0)),0),0)
  • {1,,1{1,,1♦3}1{1,,2♦2}2♦3} has recursion level C(C(Ω22+b+C(Ω2,C(Ω2+C(C(Ω22+b,0),C(Ω22,0)),0)),0),0)
  • {1,,1{1,,1♦3}1{1,,1{1,,1,,2}2♦2}2♦3} has recursion level C(C(Ω22+b+C(Ω2+C(Ω22,0),C(Ω2+C(C(Ω22+b,0),C(Ω22,0)),0)),0),0)
  • {1,,1{1,,1♦3}1{1,,1♦3}2♦3} has recursion level C(C(Ω22+b+C(Ω2+C(C(Ω22+b,0),C(Ω22,0)),C(Ω2+C(C(Ω22+b,0),C(Ω22,0)),0)),0),0)
  • {1,,1♦1,2} has recursion level C(C(Ω22+b+C(Ω2+C(C(Ω22+b,0),C(Ω22,0))+1,0),0),0)
  • {1,,1♦1{1,,1{1,,2,,2}1{1,,1,,2}2,,2}2} has recursion level C(C(Ω22+b+C(Ω2+C(C(Ω22+b,0),C(Ω22,0))+C(C(Ω22+a,0),C(Ω22,0)),0),0),0)
  • {1,,1♦1♦2} has recursion level C(C(Ω22+b+C(Ω2+C(C(Ω22+b,0),C(Ω22,0))2,0),0),0)
  • {1,,1{2,,1♦2,,2}2} has recursion level C(C(Ω22+b+C(Ω2+ω^(C(C(Ω22+b,0),C(Ω22,0))+1),0),0),0)
  • {1{1,,1,,2}2,,1♦2,,2} has recursion level C(C(Ω22+b+C(Ω2+ω^(C(C(Ω22+b,0),C(Ω22,0))+C(Ω22,0)),0),0),0)
  • {1{1,,2,,2}2{1,,1♦2,,2}2,,1,,2} has recursion level C(C(Ω22+b+C(Ω2+C(C(Ω2,C(Ω22,0)),C(C(Ω22+b,0),C(Ω22,0))),0),0),0)
  • {1{1,,1{1,,2,,2}1{1,,1,,2}2,,2}2{1,,1♦2,,2}2,,1,,2} has recursion level C(C(Ω22+b+C(Ω2+C(C(Ω22+a,0),C(C(Ω22+b,0),C(Ω22,0))),0),0),0)
  • {1{1,,1♦2,,2}3,,1,,2} has recursion level C(C(Ω22+b+C(Ω2+C(C(Ω22+b,0),C(C(Ω22+b,0),C(Ω22,0))),0),0),0)
  • {1{1,,1♦2,,2}1,2,,1,,2} has recursion level C(C(Ω22+b+C(Ω2+C(C(Ω22+b,0)+1,C(Ω22,0)),0),0),0)
  • {1{1,,1{1,,2,,2}1{1,,1,,2}2,,2}2,,1{1,,2,,2}1{1{1,,1♦2,,2}2,,1,,2}2,,2} has recursion level C(C(Ω22+b+C(Ω2+C(C(Ω22+b,0)+C(Ω22+a,0),C(Ω22,0)),0),0),0)
  • {1{1,,1{1,,2,,2}1{1{1,,1♦2,,2}2,,1,,2}2,,2}2,,1{1,,2,,2}1{1{1,,1♦2,,2}2,,1,,2}2,,2} has recursion level C(C(Ω22+b+C(Ω2+C(C(Ω22+b,0)2,C(Ω22,0)),0),0),0)
  • {1,,2{1,,2,,2}1{1{1,,1♦2,,2}2,,1,,2}2,,2} has recursion level C(C(Ω22+b+C(Ω2+C(C(Ω2,C(Ω22+b,0)),C(Ω22,0)),0),0),0)
  • {1,,1{1,,2,,2}2{1{1,,1♦2,,2}2,,1,,2}2,,2} has recursion level C(C(Ω22+b+C(Ω2+C(C(Ω22,C(Ω22+b,0)),C(Ω22,0)),0),0),0)
  • {1,,1{1,,2,,2}1{1,,1,,2}2{1{1,,1♦2,,2}2,,1,,2}2,,2} has recursion level C(C(Ω22+b+C(Ω2+C(C(Ω22+a,C(Ω22+b,0)),C(Ω22,0)),0),0),0)
  • {1,,1{1,,2,,2}1{1{1,,2,,2}2,,1,,2}2{1{1,,1♦2,,2}2,,1,,2}2,,2} has recursion level C(C(Ω22+b+C(Ω2+C(C(Ω22+C(C(Ω2,a),a),C(Ω22+b,0)),C(Ω22,0)),0),0),0)
  • {1,,1{1,,2,,2}1{1{1,,1{1,,1,,3}2,,2}2,,1,,2}2{1{1,,1♦2,,2}2,,1,,2}2,,2} has recursion level C(C(Ω22+b+C(Ω2+C(C(Ω22+C(C(Ω22,0),a),C(Ω22+b,0)),C(Ω22,0)),0),0),0)
  • {1,,1{1,,2,,2}1{1{1,,1{1,,1{1,,1,,3}2,,2}1,2{1,,1,,3}2,,2}2,,1,,2}2{1{1,,1♦2,,2}2,,1,,2}2,,2} has recursion level C(C(Ω22+b+C(Ω2+C(C(Ω22+C(C(Ω22+1,0),a),C(Ω22+b,0)),C(Ω22,0)),0),0),0)
  • {1,,1{1,,2,,2}1{1{1,,1{1,,1{1,,1,,3}2,,2}1{1,,2,,2}2{1,,1,,3}2,,2}2,,1,,2}2{1{1,,1♦2,,2}2,,1,,2}2,,2} has recursion level C(C(Ω22+b+C(Ω2+C(C(Ω22+C(Ω2,a),C(Ω22+b,0)),C(Ω22,0)),0),0),0)
  • {1,,1{1,,2,,2}1{1{1,,1{1,,1{1,,1,,3}2,,2}1{1,,1{1,,1,,3}2,,2}2{1,,1,,3}2,,2}2,,1,,2} 2{1{1,,1♦2,,2}2,,1,,2}2,,2} has recursion level C(C(Ω22+b+C(Ω2+C(C(Ω22+C(Ω2+C(Ω22,0),a),C(Ω22+b,0)),C(Ω22,0)),0),0),0)
  • {1,,1{1,,2,,2}1{1{1,,1{1,,1,,3}1,2,,2}2,,1,,2}2{1{1,,1♦2,,2}2,,1,,2}2,,2} has recursion level C(C(Ω22+b+C(Ω2+C(C(Ω22+C(Ω2+C(Ω22,0)+1,a),C(Ω22+b,0)),C(Ω22,0)),0),0),0)
  • {1,,1{1,,2,,2}1{1{1,,1{1,,1{1,,2,,3}1,2,,3}2,,2}2,,1,,2}2{1{1,,1♦2,,2}2,,1,,2}2,,2} has recursion level C(C(Ω22+b+C(Ω2+C(C(Ω22+C(Ω2+C(C(Ω22+1,0),C(Ω22,0)),a),C(Ω22+b,0)),C(Ω22,0)),0),0),0)
  • {1,,1{1,,2,,2}1{1{1,,1♦2,,2}2,,1,,2}3,,2} has recursion level C(C(Ω22+b+C(Ω2+C(C(Ω22+b,C(Ω22+b,0)),C(Ω22,0)),0),0),0)
  • {1,,1{1,,2,,2}1{1{1,,1♦2,,2}2,,1,,2}1,2,,2} has recursion level C(C(Ω22+b+C(Ω2+C(C(Ω22+b+1,0),C(Ω22,0)),0),0),0)
  • {1,,1{1,,2,,2}1{1{1,,1♦2,,2}2,,1,,2}1{1,,1{1,,1{1,,2,,2}1{1{1,,1♦2,,2}2,,1,,2}1,2,,2}2}2,,2} has recursion level C(C(Ω22+b+C(Ω2+C(C(Ω22+b+C(Ω2+C(C(Ω22+b+1,0),C(Ω22,0)),0),0),C(Ω22,0)),0),0),0)
  • {1,,1{1,,2,,2}1{1{1,,1♦2,,2}2,,1,,2}1{1,,1,,2}2,,2} has recursion level C(C(Ω22+b+a,0),0)
  • {1,,1{1,,2,,2}1{1{1,,1♦2,,2}2,,1,,2}1{1{1,,1{1,,1,,3}2,,2}2,,1,,2}2,,2} has recursion level C(C(Ω22+b+C(C(Ω22,0),a),0),0)
  • {1,,1{1,,2,,2}1{1{1,,1♦2,,2}2,,1,,2}1{1{1,,1{1,,1{1,,1,,3}2,,2}1{1,,1,,2}2{1,,1,,3}2,,2}2,,1,,2}2,,2} has recursion level C(C(Ω22+b+C(C(Ω22+a,0),a),0),0)
  • {1,,1{1,,2,,2}1{1{1,,1♦2,,2}2,,1,,2}1 {1{1,,1{1,,1{1,,1,,3}2,,2}1{1,,1{1,,1,,3}2,,2}2{1,,1,,3}2,,2}2,,1,,2}2,,2} has recursion level C(C(Ω22+b+C(C(Ω22+C(Ω2+C(Ω22,0),a),0),a),0),0)
  • {1,,1{1,,2,,2}1{1{1,,1♦2,,2}2,,1,,2}1{1{1,,1{1,,1,,3}1,2,,2}2,,1,,2}2,,2} has recursion level C(C(Ω22+b+C(C(Ω22+C(Ω2+C(Ω22,0)+1,a),0),a),0),0)
  • {1,,1{1,,2,,2}1{1{1,,1♦2,,2}2,,1,,2}1{1{1,,1{1,,1{1,,2,,3}1,2,,3}2,,2}2,,1,,2}2,,2} has recursion level C(C(Ω22+b+C(C(Ω22+C(Ω2+C(C(Ω22+1,0),C(Ω22,0)),a),0),a),0),0)
  • {1,,1{1,,2,,2}1{1{1,,1♦2,,2}2,,1,,2}1{1{1,,1♦2,,2}2,,1,,2}2,,2} has recursion level C(C(Ω22+b+C(C(Ω22+b,0),a),0),0)
  • {1,,1{1,,2,,2}1{2{1,,1♦2,,2}2,,1,,2}2,,2} has recursion level C(C(Ω22+b+ω^(C(C(Ω22+b,0),a)+1),0),0)
  • {1,,1{1,,2,,2}1{1{1,,1♦2,,2}1,2,,1,,2}2,,2} has recursion level C(C(Ω22+b+C(C(Ω22+b,0)+1,a),0),0)
  • {1,,1{1,,2,,2}1{1{1{1,,1,2,,2}2,,1♦2,,2}2,,1,,2}2,,2} has recursion level C(C(Ω22+b+C(C(Ω22+b,0)+C(Ω2+1,a),a),0),0)
  • {1{1,,1{1,,1,,3}2,,2}2,,1♦2,,2} has recursion level C(C(Ω22+b+C(C(Ω22+b,0)+C(Ω22,0),a),0),0)
  • {1{1,,1♦2,,2}2,,1♦2,,2} has recursion level C(C(Ω22+b+C(C(Ω22+b,0)2,a),0),0)
  • {1,,2♦2,,2} has recursion level C(C(Ω22+b+C(C(Ω2,C(Ω22+b,0)),a),0),0)
  • {1,,1{1,,2,,2}2♦2,,2} has recursion level C(C(Ω22+b+C(C(Ω2+C(Ω2,a),C(Ω22+b,0)),a),0),0)
  • {1,,1{1,,1{1,,2,,2}2,,2}2♦2,,2} has recursion level C(C(Ω22+b+C(C(Ω2+C(Ω2+C(Ω2,a),a),C(Ω22+b,0)),a),0),0)
  • {1,,1{1,,1{1,,1,,3}2,,2}2♦2,,2} has recursion level C(C(Ω22+b+C(C(Ω22,C(Ω22+b,0)),a),0),0)
  • {1,,1{1,,1♦2,,2}2♦2,,2} has recursion level C(C(Ω22+b+C(C(Ω22+b,C(Ω22+b,0)),a),0),0)
  • {1,,1{1,,1♦2,,2}1,2♦2,,2} has recursion level C(C(Ω22+b+C(C(Ω22+b+1,0),a),0),0)
  • {1,,1{1,,1♦2,,2}1{1{1,,1{1,,1♦2,,2}1,2♦2,,2}2,,1,,2}2♦2,,2} has recursion level C(C(Ω22+b+C(C(Ω22+b+C(C(Ω22+b+1,0),a),0),a),0),0)
  • {1,,1{1,,1♦2,,2}1{1,,2,,2}2♦2,,2} has recursion level C(C(Ω22+b+C(Ω2,a),0),0)
  • {1,,1{1,,1♦2,,2}1{1,,1{1,,1,,3}2,,2}2♦2,,2} has recursion level C(C(Ω22+b+C(Ω2+C(Ω22,0),a),0),0)
  • {1,,1{1,,1♦2,,2}1{1,,1{1,,1,,3}1,2,,2}2♦2,,2} has recursion level C(C(Ω22+b+C(Ω2+C(Ω22,0)+1,a),0),0)
  • {1,,1{1,,1♦2,,2}1{1,,1{1,,1{1,,2,,3}1,2,,3}2,,2}2♦2,,2} has recursion level C(C(Ω22+b+C(Ω2+C(C(Ω22+1,0),C(Ω22,0)),a),0),0)
  • {1,,1{1,,1♦2,,2}1{1,,1♦2,,2}2♦2,,2} has recursion level C(C(Ω22+b+C(Ω2+C(C(Ω22+b,0),C(Ω22,0)),a),0),0)
  • {1,,1{1,,1,,3}1,2♦2,,2} has recursion level C(C(Ω22+b+C(Ω2+C(Ω22,0)+1,C(Ω2+C(C(Ω22+b,0),C(Ω22,0)),a)),0),0)
  • {1,,1{1,,1{1,,2,,3}1,2,,3}2♦2,,2} has recursion level C(C(Ω22+b+C(Ω2+C(C(Ω22+1,0),C(Ω22,0)),C(Ω2+C(C(Ω22+b,0),C(Ω22,0)),a)),0),0)
  • {1,,1♦3,,2} has recursion level C(C(Ω22+b+C(Ω2+C(C(Ω22+b,0),C(Ω22,0)),C(Ω2+C(C(Ω22+b,0),C(Ω22,0)),a)),0),0)
  • {1,,1♦1,2,,2} has recursion level C(C(Ω22+b+C(Ω2+C(C(Ω22+b,0),C(Ω22,0))+1,a),0),0)
  • {1,,1{2,,1{1,,2,,3}1{1,,1,,3}2,,3}2,,2} has recursion level C(C(Ω22+b+C(Ω2+ω^(C(C(Ω22+b,0),C(Ω22,0))+1),a),0),0)
  • {1{1,,2,,3}2♦2,,1,,3} has recursion level C(C(Ω22+b+C(Ω2+C(C(Ω2,C(Ω22,0)),C(C(Ω22+b,0),C(Ω22,0))),a),0),0)
  • {1♦1,2,,1,,3} has recursion level C(C(Ω22+b+C(Ω2+C(C(Ω22+b,0)+1,C(Ω22,0)),a),0),0)
  • {1♦2,,1{1,,2,,3}1{1,,1,,3}2,,3} has recursion level C(C(Ω22+b+C(Ω2+C(C(Ω22+b,0)2,C(Ω22,0)),a),0),0)
  • {1,,2{1,,2,,3}1{1,,1,,3}2,,3} has recursion level C(C(Ω22+b+C(Ω2+C(C(Ω2,C(Ω22+b,0)),C(Ω22,0)),a),0),0)
  • {1,,1{1,,2,,3}1,2{1,,1,,3}2,,3} has recursion level C(C(Ω22+b+C(Ω2+C(C(Ω22+1,C(Ω22+b,0)),C(Ω22,0)),a),0),0)
  • {1,,1{1,,2,,3}1{1,,1,,3}3,,3} has recursion level C(C(Ω22+b+C(Ω2+C(C(Ω22+b,C(Ω22+b,0)),C(Ω22,0)),a),0),0)
  • {1,,1{1,,2,,3}1{1,,1,,3}1,2,,3} has recursion level C(C(Ω22+b+C(Ω2+C(C(Ω22+b+1,0),C(Ω22,0)),a),0),0)
  • {1,,1{1,,2,,3}1{1,,1,,3}1{1,,1,,3}2,,3} has recursion level C(C(Ω22+b2,0),0)
  • {1,,1{1,,2,,3}1{1{1,,2,,3}2,,1,,3}2,,3} has recursion level C(C(Ω22+C(C(Ω2,b),b),0),0)
  • {1,,1{1,,2,,3}1{1{1,,1{1,,1,,3}2,,3}2,,1,,3}2,,3} has recursion level C(C(Ω22+C(C(Ω2+b,b),b),0),0)
  • {1,,1{1,,2,,3}1{1{1,,1{1,,2,,3}2,,3}2,,1,,3}2,,3} has recursion level C(C(Ω22+C(C(Ω2+C(Ω2,b),b),b),0),0)
  • {1,,1{1,,2,,3}1{1,,2,,3}2,,3} has recursion level C(C(Ω22+C(C(Ω2+C(Ω2,b)2,b),b),0),0)
  • {1,,1{1,,1{1,,1,,3}2,,3}2,,3} has recursion level C(C(Ω22+C(C(Ω2+C(Ω2+b,b),b),b),0),0)
  • {1,,1,,4} has recursion level C(C(Ω22+C(C(Ω22,0),b),0),0)

And so on, the recursion level of separator {1,,1,,k} (k > 2) is C(C(Ω22+C(C(Ω22,0),αk-2),0),0), where α0 = 0 and αk+1 = C(Ω2+C(Ω2,C(Ω22,0)),αk), between C(C(Ω22+αk-2,0),0) and C(C(Ω22+αk-1,0),0). So {1,,1,,1,2} has recursion level C(C(Ω22+C(Ω2+C(Ω2,C(Ω22,0))+1,0),0),0). Now let ordinal d = C(Ω2,C(Ω22,0)).

  • {1,,1,,1,3} has recursion level C(C(Ω22+C(Ω2+d+1,C(Ω2+d+1,0)),0),0)
  • {1,,1,,1,1,2} has recursion level C(C(Ω22+C(Ω2+d+2,0),0),0)
  • {1,,1,,1{1,,1{1,,1,,2}2}2} has recursion level C(C(Ω22+C(Ω2+d+C(Ω2+C(Ω22,0),0),0),0),0)
  • {1,,1,,1{1,,1{1,,1,,2}1,2}2} has recursion level C(C(Ω22+C(Ω2+d+C(Ω2+C(Ω22,0)+1,0),0),0),0)
  • {1,,1,,1{1,,1{1,,1{1,,2,,2}1,2,,2}2}2} has recursion level C(C(Ω22+C(Ω2+d+C(Ω2+C(C(Ω22+1,0),C(Ω22,0)),0),0),0),0)
  • {1,,1,,1{1,,1{1,,1{1,,2,,2}1{1,,1,,2}2,,2}2}2} has recursion level C(C(Ω22+C(Ω2+d+C(Ω2+C(C(Ω22+C(Ω2+d,0),0),C(Ω22,0)),0),0),0),0)
  • {1,,1,,1{1,,1{1,,1,,1,2}2}2} has recursion level C(C(Ω22+C(Ω2+d+C(Ω2+C(C(Ω22+C(Ω2+d+1,0),0),C(Ω22,0)),0),0),0),0)
  • {1,,1,,1{1,,1{1,,1,,1{1,,1{1,,1,,1,2}2}2}2}2} has recursion level C(C(Ω22+C(Ω2+d+C(Ω2+C(C(Ω22+C(Ω2+d+C(Ω2+C(C(Ω22+C(Ω2+d+1,0),0),C(Ω22,0)),0),0),0),C(Ω22,0)),0),0),0),0)
  • {1,,1,,1{1,,1,,2}2} has recursion level C(C(Ω22+C(Ω2+d+C(Ω2+d,0),0),0),0)
  • {1,,1,,1,2{1,,1,,2}2} has recursion level C(C(Ω22+C(Ω2+d+1,C(Ω2+d+C(Ω2+d,0),0)),0),0)
  • {1,,1,,1{1,,1{1,,1{1,,2,,2}1{1,,1,,2}2,,2}2}2{1,,1,,2}2} has recursion level C(C(Ω22+C(Ω2+d+C(Ω2+C(C(Ω22+1,0),C(Ω22,0)),0),C(Ω2+d+C(Ω2+d,0),0)),0),0)
  • {1,,1,,1{1,,1{1,,1,,1,2}2}2{1,,1,,2}2} has recursion level C(C(Ω22+C(Ω2+d+C(Ω2+C(C(Ω22+C(Ω2+d+1,0),0),C(Ω22,0)),0),C(Ω2+d+C(Ω2+d,0),0)),0),0)
  • {1,,1,,1{1,,1{1,,1,,1{1,,1,,2}2}2}2{1,,1,,2}2} has recursion level C(C(Ω22+C(Ω2+d+C(Ω2+C(C(Ω22+C(Ω2+d+C(Ω2+d,0),0),0),C(Ω22,0)),0),C(Ω2+d+C(Ω2+d,0),0)),0),0)
  • {1,,1,,1{1,,1{1,,1,,1,2{1,,1,,2}2}2}2{1,,1,,2}2} has recursion level C(C(Ω22+C(Ω2+d+C(Ω2+C(C(Ω22+C(Ω2,C(Ω2+d+C(Ω2+d,0),0)),0),C(Ω22,0)),0),C(Ω2+d+C(Ω2+d,0),0)),0),0)
  • {1,,1,,1{1,,1,,2}3} has recursion level C(C(Ω22+C(Ω2+d+C(Ω2+d,0),C(Ω2+d+C(Ω2+d,0),0)),0),0)
  • {1,,1,,1{1,,1,,2}1,2} has recursion level C(C(Ω22+C(Ω2+d+C(Ω2+d,0)+1,0),0),0)
  • {1,,1,,1{1,,1,,2}1{1,,1,,2}2} has recursion level C(C(Ω22+C(Ω2+d+C(Ω2+d,0)2,0),0),0)
  • {1,,1,,1{1{1,,2,,2}2,,1,,2}2} has recursion level C(C(Ω22+C(Ω2+d+C(C(Ω2,C(Ω2+d,0)),C(Ω2+d,0)),0),0),0)
  • {1,,1,,1{1{1,,1,,3}2,,1,,2}2} has recursion level C(C(Ω22+C(Ω2+d+C(C(Ω22,0),C(Ω2+d,0)),0),0),0)
  • {1,,1,,1{1{1,,1,,1,2}2,,1,,2}2} has recursion level C(C(Ω22+C(Ω2+d+C(C(Ω22+C(Ω2+d+1,0),0),C(Ω2+d,0)),0),0),0)
  • {1,,1,,1{1{1,,1,,1{1{1,,1,,1,2}2,,1,,2}2}2,,1,,2}2} has recursion level C(C(Ω22+C(Ω2+d+C(C(Ω22+C(Ω2+d+C(C(Ω22+C(Ω2+d+1,0),0),C(Ω2+d,0)),0),0),C(Ω2+d,0)),0),0),0)
  • {1,,1,,1{1,,2,,2}2} has recursion level C(C(Ω22+C(Ω2+d+C(Ω2,C(Ω2+d,0)),0),0),0)
  • {1,,1,,1{1,,1{1,,1,,2}2,,2}2} has recursion level C(C(Ω22+C(Ω2+d+C(Ω2+C(Ω2+d,0),C(Ω2+d,0)),0),0),0)
  • {1,,1,,1{1,,1{1,,1,,3}2,,2}2} has recursion level C(C(Ω22+C(Ω2+d+C(Ω2+C(Ω22,0),C(Ω2+d,0)),0),0),0)
  • {1,,1,,1{1,,1{1,,1,,1,2}2,,2}2} has recursion level C(C(Ω22+C(Ω2+d+C(Ω2+C(C(Ω22+C(Ω2+d+1,0),0),C(Ω22,0)),C(Ω2+d,0)),0),0),0)
  • {1,,1,,1{1,,1{1,,1,,1{1,,1{1,,1,,1,2}2,,2}2}2,,2}2} has recursion level C(C(Ω22+C(Ω2+d+C(Ω2+C(C(Ω22+C(Ω2+d+C(Ω2+C(C(Ω22+C(Ω2+d+1,0),0),C(Ω22,0)),C(Ω2+d,0)),0),0),C(Ω22,0)),C(Ω2+d,0)),0),0),0)
  • {1,,1,,1{1,,1,,3}2} has recursion level C(C(Ω22+C(Ω2+d+C(Ω2+d,C(Ω2+d,0)),0),0),0)
  • {1,,1,,1{1,,1,,1,2}2} has recursion level C(C(Ω22+C(Ω2+d+C(Ω2+d+1,0),0),0),0)
  • {1,,1,,1{1,,1,,1{1,,1,,2}2}2} has recursion level C(C(Ω22+C(Ω2+d+C(Ω2+d+C(Ω2+d,0),0),0),0),0)
  • {1,,1,,1{1,,1,,1{1,,1,,1,2}2}2} has recursion level C(C(Ω22+C(Ω2+d+C(Ω2+d+C(Ω2+d+1,0),0),0),0),0)
  • {1,,1,,1,,2} has recursion level C(C(Ω22+C(Ω2+d+C(Ω22,0),0),0),0)

So {1,,1,,1,,2} has recursion level C(C(Ω22+C(Ω2+C(Ω2,C(Ω22,0))+C(Ω22,0),0),0),0). Here’re some approximations, to make the comparisons above more simple:

  • {1,,1,,2} approximately corresponds to C(Ω2+C(Ω2,C(Ω22,0)),0)
  • {1,,2,,2} approximately corresponds to C(Ω2,C(Ω2+C(Ω2,C(Ω22,0)),0))
  • {1,,1{1,,1,,2}2,,2} approximately corresponds to C(Ω2+C(Ω2+C(Ω2,C(Ω22,0)),0),C(Ω2+C(Ω2,C(Ω22,0)),0))
  • {1,,1{1,,1,,3}2,,2} approximately corresponds to C(Ω2+C(Ω22,0),C(Ω2+C(Ω2,C(Ω22,0)),0))
  • {1,,1{1,,1{1,,2,,3}1{1,,1,,3}2,,3}2,,2} approximately corresponds to C(Ω2+C(C(Ω22+C(Ω2+C(Ω2,C(Ω22,0)),C(Ω2+C(Ω2,C(Ω22,0)),0)),0),C(Ω22,0)),C(Ω2+C(Ω2,C(Ω22,0)),0))
  • {1,,1,,3} approximately corresponds to C(Ω2+C(Ω2,C(Ω22,0)),C(Ω2+C(Ω2,C(Ω22,0)),0))
  • {1,,2,,3} approximately corresponds to C(Ω2,C(Ω2+C(Ω2,C(Ω22,0)),C(Ω2+C(Ω2,C(Ω22,0)),0)))
  • {1,,1{1,,1,,4}2,,3} approximately corresponds to C(Ω2+C(Ω22,0),C(Ω2+C(Ω2,C(Ω22,0)),C(Ω2+C(Ω2,C(Ω22,0)),0)))
  • {1,,1,,4} approximately corresponds to C(Ω2+C(Ω2,C(Ω22,0)),C(Ω2+C(Ω2,C(Ω22,0)),C(Ω2+C(Ω2,C(Ω22,0)),0)))
  • {1,,1,,1,2} approximately corresponds to C(Ω2+C(Ω2,C(Ω22,0))+1,0)
  • {1,,1,,1{1,,1{1,,1,,2}2}2} approximately corresponds to C(Ω2+C(Ω2,C(Ω22,0))+C(Ω2+C(Ω22,0),0),0)
  • {1,,1,,1{1,,1,,2}2} approximately corresponds to C(Ω2+C(Ω2,C(Ω22,0))+C(Ω2+C(Ω2,C(Ω22,0)),0),0)

Beyond {1,,1,,1,,2}: Conclusion

Due to the outside layer of C(C(Ω22+____,0),0), the comparisons above seem erratic. But now we find the patterns on them.

The story begins from C(Ω2+C(Ω22,0),0), corresponding to {1,,1{1,,1,,2}2}. Then εC(Ω2+C(Ω22,0),0)+1 = C(C(Ω2,C(Ω2+C(Ω22,0),0)),C(Ω2+C(Ω22,0),0)) corresponds to {1{1,,2{1,,1,,2}2}2,,1{1,,1,,2}2}. When the α in C(α,C(Ω2+C(Ω22,0),0)) get larger (but α < Ω2), it’ll correspond to {1 A 2,,1{1,,1,,2}2} with higher-leveled A (here A can be very high-leveled). The least ordinal larger than all those C(α,C(Ω2+C(Ω22,0),0)) is C(Ω2,C(Ω2+C(Ω22,0),0)), so C(Ω2,C(Ω2+C(Ω22,0),0)) works as the 1-separator in C(____,C(Ω2+C(Ω22,0),0)), just like that {1,,2{1,,1,,2}2} is the 1-separator in { ____ ,,1{1,,1,,2}2}.

Next, C(Ω2,C(Ω2,C(Ω2+C(Ω22,0),0))) corresponds to {1,,3{1,,1,,2}2}, and so on. C(Ω2+C(Ω22,0),C(Ω2+C(Ω22,0),0)) corresponds to {1,,1{1,,1,,2}3}. Generally, for β < C(Ω22,0), the first fixed point of x -> C(Ω2+x,β) is C(Ω2+C(Ω22,0),β), and C(Ω22,0) works as the 1-separator in C(Ω2+____,β), just like that {1,,1,,2} works as the 1-separator in {1,, ____ }.

But this correspondence fails at the next key point – When the α in C(Ω2+C(α,C(Ω22,0)),0) get larger (but α < Ω2), it’ll correspond to {1,,1 A 2} with higher-leveled A (here A can be very high-leveled). Note that without the limit from outside layer of C(C(Ω22+____,0),0), the α can be larger than, for instance, C(Ω23,0). The least ordinal larger than all those C(Ω2+C(α,C(Ω22,0)),0) is C(Ω2+C(Ω2,C(Ω22,0)),0), so actually C(Ω2+C(Ω2,C(Ω22,0)),0) corresponds to {1,,1,,2}, the 1-separator in {1,, ____ }. But the C(Ω22,0) still works as the 1-separator in α inside C(Ω2+α,β) (α < Ω2, β < C(Ω22,0)). This property is important even in further comparisons.

Next, when the α in C(α,C(Ω2+C(Ω2,C(Ω22,0)),0)) get larger (but α < Ω2), it’ll correspond to {1 A 2,,1,,2} with higher-leveled A (here A can be very high-leveled). The least ordinal larger than all those C(α,C(Ω2+C(Ω2,C(Ω22,0)),0)) is C(Ω2,C(Ω2+C(Ω2,C(Ω22,0)),0)), so C(Ω2,C(Ω2+C(Ω2,C(Ω22,0)),0)) corresponds to {1,,2,,2}, the 1-separator in { ____ ,,1,,2}.

Next, C(Ω2+C(Ω2,C(Ω22,0)),C(Ω2+C(Ω2,C(Ω22,0)),0)) corresponds to {1,,1,,3}, the 1-separator in {1,, ____ ,,2}. Generally, for β < C(Ω22,0), when the α in C(Ω2+C(α,C(Ω22,0)),β) get larger (but α < Ω2), it can traverse very high-leveled separator A in, for instance, {1,,1…,,1,,1 A 2…}. The least ordinal larger than all those C(Ω2+C(α,C(Ω22,0)),β) is C(Ω2+C(Ω2,C(Ω22,0)),β), so C(Ω2,C(Ω22,0)) works as the 2-separator in C(Ω2+____,β), just like the double comma.

But this correspondence fails at some higher point (see the later post).

Now let separator ◊ = {1,,1,,1{1,,1,,1,,2}2} and let ordinal d = C(Ω2,C(Ω22,0)), and a = C(Ω2+d+C(Ω22,0),0) = C(Ω2+C(Ω2,C(Ω22,0))+C(Ω22,0),0).

  • {1,,1{1{1,,1◊2}2,,1{1,,1,,2}2}2{1,,1,,2}2} has recursion level C(C(Ω22+a,0),0)
  • {1,,1{2{1,,1◊2}2,,1{1,,1,,2}2}2{1,,1,,2}2} has recursion level C(C(Ω22+a+ω^(C(C(Ω22+a,0),C(Ω2+C(Ω22,0),0))+1),0),0)
  • {1,,1{1{1,,1◊2}1,2,,1{1,,1,,2}2}2{1,,1,,2}2} has recursion level C(C(Ω22+a+C(C(Ω22+a,0)+1,C(Ω2+C(Ω22,0),0)),0),0)
  • {1{1,,1{1,,1,,2}1,2}2,,1◊2} has recursion level C(C(Ω22+a+C(C(Ω22+a,0)+C(Ω22+C(Ω2+C(Ω22,0)+1,0),0),C(Ω2+C(Ω22,0),0)),0),0)
  • {1{1,,1{1,,1{1,,2,,2}1{1,,1,,2}2,,2}2}2,,1◊2} has recursion level C(C(Ω22+a+C(C(Ω22+a,0)+C(Ω22+C(Ω2+d,0),0),C(Ω2+C(Ω22,0),0)),0),0)
  • {1{1,,1◊2}2,,1◊2} has recursion level C(C(Ω22+a+C(C(Ω22+a,0)2,C(Ω2+C(Ω22,0),0)),0),0)
  • {1,,2◊2} has recursion level C(C(Ω22+a+C(C(Ω2,C(Ω22+a,0)),C(Ω2+C(Ω22,0),0)),0),0)
  • {1,,1{1,,1◊2}2◊2} has recursion level C(C(Ω22+a+C(C(Ω22+a,C(Ω22+a,0)),C(Ω2+C(Ω22,0),0)),0),0)
  • {1,,1{1,,1◊2}1,2◊2} has recursion level C(C(Ω22+a+C(C(Ω22+a+1,0),C(Ω2+C(Ω22,0),0)),0),0)
  • {1,,1{1,,1◊2}1{1,,2{1,,1,,2}2}2◊2} has recursion level C(C(Ω22+a+C(Ω2,C(Ω2+C(Ω22,0),0)),0),0)
  • {1,,1{1,,1◊2}1{1,,1{1,,1,,2}1,2}2◊2} has recursion level C(C(Ω22+a+C(Ω2+C(Ω22,0)+1,0),0),0)
  • {1,,1{1,,1◊2}1{1,,1{1,,1{1,,2,,2}1{1,,1,,2}2,,2}2}2◊2} has recursion level C(C(Ω22+a+C(Ω2+C(C(Ω22+C(Ω2+d,0),0),C(Ω22,0)),0),0),0)
  • {1,,1{1,,1◊2}1{1,,1◊2}2◊2} has recursion level C(C(Ω22+a+C(Ω2+C(C(Ω22+a,0),C(Ω22,0)),0),0),0)
  • {1,,1{1,,1,,2}2◊2} has recursion level C(C(Ω22+a+C(C(Ω22,0),C(Ω2+C(C(Ω22+a,0),C(Ω22,0)),0)),0),0)
  • {1,,1{1,,1{1,,1,,2}2◊2}1{1,,1◊2}2{1,,1,,2}2◊2} has recursion level C(C(Ω22+a+C(C(Ω22,0),C(Ω2+C(C(Ω22+a,0),C(Ω22,0)),0))+C(Ω2+C(C(Ω22+a,0),C(Ω22,0)),0),0),0)
  • {1,,1{1,,1{1,,1,,2}2◊2}1{1{1,,1{1,,1,,2}2◊2}2,,1◊2}2{1,,1,,2}2◊2} has recursion level C(C(Ω22+a+C(C(Ω22,0),C(Ω2+C(C(Ω22+a,0),C(Ω22,0)),0))2,0),0)
  • {1,,1{1,,1{1,,1,,2}2◊2}1{1{1,,2{1,,1,,2}2◊2}2,,1◊2}2{1,,1,,2}2◊2} has recursion level C(C(Ω22+a+C(C(Ω2,C(Ω22,0)),C(Ω2+C(C(Ω22+a,0),C(Ω22,0)),0)),0),0)
  • {1,,1{1,,1{1,,1,,2}2◊2}1{1{1,,1{1,,1{1,,1,,2}2◊2}2{1,,1,,2}2◊2}2,,1◊2}2{1,,1,,2}2◊2} has recursion level C(C(Ω22+a+C(C(Ω22,C(Ω22,0)),C(Ω2+C(C(Ω22+a,0),C(Ω22,0)),0)),0),0)
  • {1,,1{1,,1{1,,1,,2}2◊2}1{1{1,,1{1,,1{1,,1,,2}2◊2}1,2{1,,1,,2}2◊2}2,,1◊2}2{1,,1,,2}2◊2} has recursion level C(C(Ω22+a+C(C(Ω22+1,0),C(Ω2+C(C(Ω22+a,0),C(Ω22,0)),0)),0),0)
  • {1,,1{1,,1{1,,1,,2}2◊2}1{1,,2◊2}2{1,,1,,2}2◊2} has recursion level C(C(Ω22+a+C(C(Ω22+C(Ω2,C(Ω2+C(C(Ω22+a,0),C(Ω22,0)),0)),0),C(Ω2+C(C(Ω22+a,0),C(Ω22,0)),0)),0),0)
  • {1,,1{1,,1{1,,1,,2}2◊2}1{1,,1{1,,1,,2}2◊2}2{1,,1,,2}2◊2} has recursion level C(C(Ω22+a+C(C(Ω22+C(Ω2+C(Ω22,0),C(Ω2+C(C(Ω22+a,0),C(Ω22,0)),0)),0),C(Ω2+C(C(Ω22+a,0),C(Ω22,0)),0)),0),0)
  • {1,,1{1,,1,,2}1,2◊2} has recursion level C(C(Ω22+a+C(C(Ω22+C(Ω2+C(Ω22,0)+1,C(Ω2+C(C(Ω22+a,0),C(Ω22,0)),0)),0),C(Ω2+C(C(Ω22+a,0),C(Ω22,0)),0)),0),0)
  • {1,,1{1,,1{1,,2,,2}1,2,,2}2◊2} has recursion level C(C(Ω22+a+C(C(Ω22+C(Ω2+C(C(Ω22+1,0),C(Ω22,0)),C(Ω2+C(C(Ω22+a,0),C(Ω22,0)),0)),0),C(Ω2+C(C(Ω22+a,0),C(Ω22,0)),0)),0),0)
  • {1,,1{1,,1{1,,2,,2}1{1,,1,,2}2,,2}2◊2} has recursion level C(C(Ω22+a+C(C(Ω22+C(Ω2+d,0),0),C(Ω2+C(C(Ω22+a,0),C(Ω22,0)),0)),0),0)
  • {1,,1{1,,1,,1,2}2◊2} has recursion level C(C(Ω22+a+C(C(Ω22+C(Ω2+d+1,0),0),C(Ω2+C(C(Ω22+a,0),C(Ω22,0)),0)),0),0)
  • {1,,1◊3} has recursion level C(C(Ω22+a+C(C(Ω22+a,0),C(Ω2+C(C(Ω22+a,0),C(Ω22,0)),0)),0),0)
  • {1,,1{1,,1◊3}1{1,,1◊2}2◊3} has recursion level C(C(Ω22+a+C(C(Ω22+a,0),C(Ω2+C(C(Ω22+a,0),C(Ω22,0)),0))+C(Ω2+C(C(Ω22+a,0),C(Ω22,0)),0),0),0)
  • {1,,1{1,,1◊3}1{1{1,,1◊3}2,,1◊2}2◊3} has recursion level C(C(Ω22+a+C(C(Ω22+a,0),C(Ω2+C(C(Ω22+a,0),C(Ω22,0)),0))2,0),0)
  • {1,,1{1,,1◊3}1{1{1,,1◊3}1,2,,1◊2}2◊3} has recursion level C(C(Ω22+a+C(C(Ω22+a,0)+1,C(Ω2+C(C(Ω22+a,0),C(Ω22,0)),0)),0),0)
  • {1,,1{1,,1◊3}1{1{1{1,,1{1,,1,,2}2◊2}2,,1◊3}2,,1◊2}2◊3} has recursion level C(C(Ω22+a+C(C(Ω22+a,0)+C(Ω22,0),C(Ω2+C(C(Ω22+a,0),C(Ω22,0)),0)),0),0)
  • {1,,1{1,,1◊3}1{1{1{1,,1◊3}2,,1◊3}2,,1◊2}2◊3} has recursion level C(C(Ω22+a+C(C(Ω22+a,0)2,C(Ω2+C(C(Ω22+a,0),C(Ω22,0)),0)),0),0)
  • {1,,1{1,,1◊3}1{1{1,,2◊3}2,,1◊2}2◊3} has recursion level C(C(Ω22+a+C(C(Ω2,C(Ω22+a,0)),C(Ω2+C(C(Ω22+a,0),C(Ω22,0)),0)),0),0)
  • {1,,1{1,,1◊3}1{1{1,,1{1,,1◊3}1,2◊3}2,,1◊2}2◊3} has recursion level C(C(Ω22+a+C(C(Ω22+a+1,0),C(Ω2+C(C(Ω22+a,0),C(Ω22,0)),0)),0),0)
  • {1,,1{1,,1◊3}1{1{1,,1{1,,1◊3}1{1,,1◊2}2◊3}2,,1◊2}2◊3} has recursion level C(C(Ω22+a+C(C(Ω22+a+C(Ω2+C(C(Ω22+a,0),C(Ω22,0)),0),0),C(Ω2+C(C(Ω22+a,0),C(Ω22,0)),0)),0),0)
  • {1,,1{1,,1◊3}1{1,,2◊2}2◊3} has recursion level C(C(Ω22+a+C(Ω2,C(Ω2+C(C(Ω22+a,0),C(Ω22,0)),0)),0),0)
  • {1,,1{1,,1◊3}1{1,,1{1,,1,,2}2◊2}2◊3} has recursion level C(C(Ω22+a+C(Ω2+C(Ω22,0),C(Ω2+C(C(Ω22+a,0),C(Ω22,0)),0)),0),0)
  • {1,,1{1,,1◊3}1{1,,1◊3}2◊3} has recursion level C(C(Ω22+a+C(Ω2+C(C(Ω22+a,0),C(Ω22,0)),C(Ω2+C(C(Ω22+a,0),C(Ω22,0)),0)),0),0)
  • {1,,1◊1,2} has recursion level C(C(Ω22+a+C(Ω2+C(C(Ω22+a,0),C(Ω22,0))+1,0),0),0)
  • {1,,1◊1◊2} has recursion level C(C(Ω22+a+C(Ω2+C(C(Ω22+a,0),C(Ω22,0))2,0),0),0)
  • {1,,1{2,,1◊2,,2}2} has recursion level C(C(Ω22+a+C(Ω2+ω^(C(C(Ω22+a,0),C(Ω22,0))+1),0),0),0)
  • {1{1,,1,,2}2,,1,,1{1,,1,,1,,2}2} has recursion level C(C(Ω22+a+C(Ω2+ω^(C(C(Ω22+a,0),C(Ω22,0))+C(Ω22,0)),0),0),0)
  • {1{1,,1{1,,2,,2}1,2,,2}2{1,,1◊2,,2}2,,1,,2} has recursion level C(C(Ω22+a+C(Ω2+C(C(Ω22+1,0),C(C(Ω22+a,0),C(Ω22,0))),0),0),0)
  • {1{1,,1◊2,,2}1,2,,1,,2} has recursion level C(C(Ω22+a+C(Ω2+C(C(Ω22+a,0)+1,C(Ω22,0)),0),0),0)
  • {1{1,,1{1,,2,,2}1{1{1,,1◊2,,2}2,,1,,2}2,,2}2,,1{1,,2,,2}1{1{1,,1◊2,,2}2,,1,,2}2,,2} has recursion level C(C(Ω22+a+C(Ω2+C(C(Ω22+a,0)2,C(Ω22,0)),0),0),0)
  • {1,,2{1,,2,,2}1{1{1,,1◊2,,2}2,,1,,2}2,,2} has recursion level C(C(Ω22+a+C(Ω2+C(C(Ω2,C(Ω22+a,0)),C(Ω22,0)),0),0),0)
  • {1,,1{1,,2,,2}2{1{1,,1◊2,,2}2,,1,,2}2,,2} has recursion level C(C(Ω22+a+C(Ω2+C(C(Ω22,C(Ω22+a,0)),C(Ω22,0)),0),0),0)
  • {1,,1{1,,2,,2}1{1,,1,,2}2{1{1,,1◊2,,2}2,,1,,2}2,,2} has recursion level C(C(Ω22+a+C(Ω2+C(C(Ω22+C(Ω2+d,0),C(Ω22+a,0)),C(Ω22,0)),0),0),0)
  • {1,,1{1,,2,,2}1{1{1,,1{1,,1,,3}2,,2}2,,1,,2}2{1{1,,1◊2,,2}2,,1,,2}2,,2} has recursion level C(C(Ω22+a+C(Ω2+C(C(Ω22+C(C(Ω22,0),C(Ω2+d,0)),C(Ω22+a,0)),C(Ω22,0)),0),0),0)
  • {1,,1{1,,2,,2}1{1{1,,1{1,,1,,1,2}2,,2}2,,1,,2}2{1{1,,1◊2,,2}2,,1,,2}2,,2} has recursion level C(C(Ω22+a+C(Ω2+C(C(Ω22+C(Ω2+d+1,0),C(Ω22+a,0)),C(Ω22,0)),0),0),0)
  • {1,,1{1,,2,,2}1{1{1,,1◊2,,2}2,,1,,2}3,,2} has recursion level C(C(Ω22+a+C(Ω2+C(C(Ω22+a,C(Ω22+a,0)),C(Ω22,0)),0),0),0)
  • {1,,1{1,,2,,2}1{1{1,,1◊2,,2}2,,1,,2}1,2,,2} has recursion level C(C(Ω22+a+C(Ω2+C(C(Ω22+a+1,0),C(Ω22,0)),0),0),0)
  • {1,,1{1,,2,,2}1{1{1,,1◊2,,2}2,,1,,2}1{1,,1,,2}2,,2} has recursion level C(C(Ω22+a+C(Ω2+d,0),0),0)
  • {1,,1{1,,2,,2}1{1{1,,1◊2,,2}2,,1,,2}1{1{1,,1{1,,1,,3}2,,2}2,,1,,2}2,,2} has recursion level C(C(Ω22+a+C(C(Ω22,0),C(Ω2+d,0)),0),0)
  • {1,,1{1,,2,,2}1{1{1,,1◊2,,2}2,,1,,2}1{1{1,,1◊2,,2}2,,1,,2}2,,2} has recursion level C(C(Ω22+a+C(C(Ω22+a,0),C(Ω2+d,0)),0),0)
  • {1,,1{1,,2,,2}1{1{1,,1◊2,,2}1,2,,1,,2}2,,2} has recursion level C(C(Ω22+a+C(C(Ω22+a,0)+1,C(Ω2+d,0)),0),0)
  • {1,,2◊2,,2} has recursion level C(C(Ω22+a+C(C(Ω2,C(Ω22+a,0)),C(Ω2+d,0)),0),0)
  • {1,,1{1,,2,,2}2◊2,,2} has recursion level C(C(Ω22+a+C(C(Ω2+C(Ω2,C(Ω2+d,0)),C(Ω22+a,0)),C(Ω2+d,0)),0),0)
  • {1,,1{1,,1{1,,1,,3}2,,2}2◊2,,2} has recursion level C(C(Ω22+a+C(C(Ω22,C(Ω22+a,0)),C(Ω2+d,0)),0),0)
  • {1,,1{1,,1◊2,,2}2◊2,,2} has recursion level C(C(Ω22+a+C(C(Ω22+a,C(Ω22+a,0)),C(Ω2+d,0)),0),0)
  • {1,,1{1,,1◊2,,2}1,2◊2,,2} has recursion level C(C(Ω22+a+C(C(Ω22+a+1,0),C(Ω2+d,0)),0),0)
  • {1,,1{1,,1◊2,,2}1{1,,2,,2}2◊2,,2} has recursion level C(C(Ω22+a+C(Ω2,C(Ω2+d,0)),0),0)
  • {1,,1{1,,1◊2,,2}1{1,,1{1,,1,,3}2,,2}2◊2,,2} has recursion level C(C(Ω22+a+C(Ω2+C(Ω22,0),C(Ω2+d,0)),0),0)
  • {1,,1{1,,1◊2,,2}1{1,,1◊2,,2}2◊2,,2} has recursion level C(C(Ω22+a+C(Ω2+C(C(Ω22+a,0),C(Ω22,0)),C(Ω2+d,0)),0),0)
  • {1,,1◊1,2,,2} has recursion level C(C(Ω22+a+C(Ω2+C(C(Ω22+a,0),C(Ω22,0))+1,C(Ω2+d,0)),0),0)
  • {1{1,,1,,3}2,,1,,1{1,,1,,1,,2}2} has recursion level C(C(Ω22+a+C(Ω2+ω^(C(C(Ω22+a,0),C(Ω22,0))+C(Ω22,0)),C(Ω2+d,0)),0),0)
  • {1{1,,1,,4}2,,1,,1{1,,1,,1,,2}2} has recursion level C(C(Ω22+a+C(Ω2+ω^(C(C(Ω22+a,0),C(Ω22,0))+C(Ω22,0)),C(Ω2+d,C(Ω2+d,0))),0),0)
  • {1{1,,1,,1,2}2,,1,,1{1,,1,,1,,2}2} has recursion level C(C(Ω22+a+C(Ω2+d+1,0),0),0)
  • {1{1,,1,,1{1,,1,,2}2}2,,1,,1{1,,1,,1,,2}2} has recursion level C(C(Ω22+a+C(Ω2+d+C(Ω2+d,0),0),0),0)
  • {1◊2,,1,,1{1,,1,,1,,2}2} has recursion level C(C(Ω22+a2,0),0)
  • {1,,2,,1{1,,1,,1,,2}2} has recursion level C(C(Ω22+C(C(Ω2,a),a),0),0)
  • {1,,1,,2{1,,1,,1,,2}2} has recursion level C(C(Ω22+C(C(Ω22,0),a),0),0)
  • {1,,1{1,,1,,2{1,,1,,1,,2}2}1,2,,1{1,,1,,1,,2}2} has recursion level C(C(Ω22+C(Ω2+C(Ω22,0)+1,a),0),0)
  • {1,,1{1,,2,,2{1,,1,,1,,2}2}1,2,,2{1,,1,,1,,2}2} has recursion level C(C(Ω22+C(Ω2+C(C(Ω22+1,0),C(Ω22,0)),a),0),0)
  • {1,,1{1,,2,,2{1,,1,,1,,2}2}1{1,,1,,2}2,,2{1,,1,,1,,2}2} has recursion level C(C(Ω22+C(Ω2+C(C(Ω22+C(Ω2+d,0),0),C(Ω22,0)),a),0),0)
  • {1,,1{1,,2,,2{1,,1,,1,,2}2}1◊2,,2{1,,1,,1,,2}2} has recursion level C(C(Ω22+C(Ω2+C(C(Ω22+a,0),C(Ω22,0)),a),0),0)
  • {1,,1{1,,2,,2{1,,1,,1,,2}2}1{1,,2,,1{1,,1,,1,,2}2}2,,2{1,,1,,1,,2}2} has recursion level C(C(Ω22+C(Ω2+C(C(Ω22+C(Ω2,a),0),C(Ω22,0)),a),0),0)
  • {1,,1{1,,2,,2{1,,1,,1,,2}2}1{1,,1{1,,1,,2{1,,1,,1,,2}2}1,2,,1{1,,1,,1,,2}2}2,,2{1,,1,,1,,2}2} has recursion level C(C(Ω22+C(Ω2+C(C(Ω22+C(Ω2+C(Ω22,0)+1,a),0),C(Ω22,0)),a),0),0)
  • {1,,1{1,,2,,2{1,,1,,1,,2}2}1{1,,1,,2{1,,1,,1,,2}2}2,,2{1,,1,,1,,2}2} has recursion level C(C(Ω22+C(Ω2+d,a),0),0)
  • {1,,1,,1,2{1,,1,,1,,2}2} has recursion level C(C(Ω22+C(Ω2+d+1,a),0),0)
  • {1,,1,,1◊2{1,,1,,1,,2}2} has recursion level C(C(Ω22+C(Ω2+d+a,a),0),0)
  • {1,,1,,1{1,,1,,1,,2}3} has recursion level C(C(Ω22+C(Ω2+d+C(Ω22,0),C(Ω2+d+C(Ω22,0),0)),0),0)
  • {1,,1,,1{1,,1,,1,,2}4} has recursion level C(C(Ω22+C(Ω2+d+C(Ω22,0),C(Ω2+d+C(Ω22,0),C(Ω2+d+C(Ω22,0),0))),0),0)
  • {1,,1,,1{1,,1,,1,,2}1,2} has recursion level C(C(Ω22+C(Ω2+d+C(Ω22,0)+1,0),0),0)

Now let ordinal d = C(Ω2,C(Ω22,0)).

  • {1,,1,,1{1,,1,,1,,2}1{1,,1,,1,,2}2} has recursion level C(C(Ω22+C(Ω2+d+C(Ω22,0)2,0),0),0)
  • {1,,1,,1{2,,1,,1,,2}2} has recursion level C(C(Ω22+C(Ω2+d+ω^(C(Ω22,0)+1),0),0),0)
  • {1,,1,,1{1{1,,1,,1,,2}2,,1,,1,,2}2} has recursion level C(C(Ω22+C(Ω2+d+ω^(C(Ω22,0)2),0),0),0)
  • {1,,2,,1,,2} has recursion level C(C(Ω22+C(Ω2+d+C(d,C(Ω22,0)),0),0),0)
  • {1,,1,2,,1,,2} has recursion level C(C(Ω22+C(Ω2+d+C(C(Ω2+1,C(Ω22,0)),C(Ω22,0)),0),0),0)
  • {1,,1{1,,1,,1,,2}2,,1,,2} has recursion level C(C(Ω22+C(Ω2+d+C(C(Ω2+C(Ω22,0),C(Ω22,0)),C(Ω22,0)),0),0),0)
  • {1,,1{1{1,,2,,1,,2}2,,1,,1,,2}2,,1,,2} has recursion level C(C(Ω22+C(Ω2+d+C(C(Ω2+C(d,C(Ω22,0)),C(Ω22,0)),C(Ω22,0)),0),0),0)
  • {1,,1{1{1,,1{1,,1,,1,,2}2,,1,,2}2,,1,,1,,2}2,,1,,2} has recursion level C(C(Ω22+C(Ω2+d+C(C(Ω2+C(C(Ω2+C(Ω22,0),C(Ω22,0)),C(Ω22,0)),C(Ω22,0)),C(Ω22,0)),0),0),0)
  • {1,,1{1,,2,,1,,2}2,,1,,2} has recursion level C(C(Ω22+C(Ω2+d+C(C(Ω2+d,C(Ω22,0)),C(Ω22,0)),0),0),0) (here’s a new irregular thing)
  • {1,,1{1,,2,,1,,2}1,2,,1,,2} has recursion level C(C(Ω22+C(Ω2+d+C(C(Ω2+d+1,C(Ω22,0)),C(Ω22,0)),0),0),0)
  • {1,,1{1,,2,,1,,2}1{1,,1,,1,,2}2,,1,,2} has recursion level C(C(Ω22+C(Ω2+d+C(C(Ω2+d+C(Ω22,0),C(Ω22,0)),C(Ω22,0)),0),0),0)
  • {1,,1{1,,2,,1,,2}1{1,,2,,1,,2}2,,1,,2} has recursion level C(C(Ω22+C(Ω2+d+C(C(Ω22,C(Ω22,0)),C(Ω22,0)),0),0),0)
  • {1,,1{1,,2,,1,,2}1{1,,2,,1,,2}1,2,,1,,2} has recursion level C(C(Ω22+C(Ω2+d+C(C(Ω22+1,0),C(Ω22,0)),0),0),0)
  • {1,,1{1,,2,,1,,2}1{1,,2,,1,,2}1{1,,1,,1{1,,1,,1,,2}2}2,,1,,2} has recursion level C(C(Ω22+C(Ω2+d+C(C(Ω22+C(Ω2+d+C(Ω22,0),0),0),C(Ω22,0)),0),0),0)
  • {1,,1{1,,2,,1,,2}1{1,,2,,1,,2}1{1,,1,,1,,2}2,,1,,2} has recursion level C(C(Ω22+C(Ω2+d2,0),0),0)
  • {1,,1{1,,2,,1,,2}1{1,,2,,1,,2}1{1,,1,,1,,2}1{1,,1,,1,,2}2,,1,,2} has recursion level C(C(Ω22+C(Ω2+d2,0)2,0),0)
  • {1,,1{1,,2,,1,,2}1{1,,2,,1,,2}1{1,,2,,1,,2}2,,1,,2} has recursion level C(C(Ω22+C(C(Ω2+C(Ω2,C(Ω2+d2,0))3,C(Ω2+d2,0)),C(Ω2+d2,0)),0),0)
  • {1,,1,,2,,2} has recursion level C(C(Ω22+C(C(Ω22,0),C(Ω2+d2,0)),0),0)
  • {1,,1{1,,1{1,,1,,2,,2}2,,1,,2}1,2{1,,1,,2,,2}2,,1,,2} has recursion level C(C(Ω22+C(C(Ω22+1,0),C(Ω2+d2,0)),0),0)
  • {1,,1{1,,1,,2,,2}1,2,,1,,2} has recursion level C(C(Ω22+C(Ω2+C(Ω22,0)+1,C(Ω2+d2,0)),0),0)
  • {1,,1{1,,2,,2,,2}1{1,,1,,2,,2}2,,2,,2} has recursion level C(C(Ω22+C(Ω2+d,C(Ω2+d2,0)),0),0)
  • {1,,1{1,,2,,3,,2}1{1,,1,,3,,2}2,,3,,2} has recursion level C(C(Ω22+C(Ω2+d,C(Ω2+d,C(Ω2+d2,0))),0),0)
  • {1,,1,,1,2,,2} has recursion level C(C(Ω22+C(Ω2+d+1,C(Ω2+d2,0)),0),0)
  • {1,,1,,1{1,,1,,2}2,,2} has recursion level C(C(Ω22+C(Ω2+d+C(Ω2+d,0),C(Ω2+d2,0)),0),0)
  • {1,,1,,1{1,,1,,1,,2}2,,2} has recursion level C(C(Ω22+C(Ω2+d+C(Ω2+d2,0),C(Ω2+d2,0)),0),0)
  • {1,,1,,1{1,,1,,1,2,,2}2,,2} has recursion level C(C(Ω22+C(Ω2+d+C(Ω2+d+1,C(Ω2+d2,0)),C(Ω2+d2,0)),0),0)
  • {1,,1,,1,,3} has recursion level C(C(Ω22+C(Ω2+d+C(Ω22,0),C(Ω2+d2,0)),0),0)
  • {1,,1,,1{1,,1,,1,,3}1,2,,2} has recursion level C(C(Ω22+C(Ω2+d+C(Ω22,0)+1,C(Ω2+d2,0)),0),0)
  • {1,,2,,1,,3} has recursion level C(C(Ω22+C(Ω2+d+C(d,C(Ω22,0)),C(Ω2+d2,0)),0),0)
  • {1,,1{1,,2,,1,,3}2,,1,,3} has recursion level C(C(Ω22+C(Ω2+d+C(C(Ω2+d,C(Ω22,0)),C(Ω22,0)),C(Ω2+d2,0)),0),0)
  • {1,,1{1,,2,,1,,3}1{1,,2,,1,,3}2,,1,,3} has recursion level C(C(Ω22+C(Ω2+d+C(C(Ω22,C(Ω22,0)),C(Ω22,0)),C(Ω2+d2,0)),0),0)
  • {1,,1{1,,2,,1,,3}1{1,,2,,1,,3}1{1,,1,,1,,3}2,,1,,3} has recursion level C(C(Ω22+C(Ω2+d2,C(Ω2+d2,0)),0),0)
  • {1,,1,,1,,4} has recursion level C(C(Ω22+C(Ω2+d+C(Ω22,0),C(Ω2+d2,C(Ω2+d2,0))),0),0)
  • {1,,1{1,,2,,1,,4}1{1,,2,,1,,4}1{1,,1,,1,,4}2,,1,,4} has recursion level C(C(Ω22+C(Ω2+d2,C(Ω2+d2,C(Ω2+d2,0))),0),0)
  • {1,,1,,1,,1,2} has recursion level C(C(Ω22+C(Ω2+d2+1,0),0),0)
  • {1,,1,,1,,1{1,,1,,1,,2}2} has recursion level C(C(Ω22+C(Ω2+d2+C(Ω2+d2,0),0),0),0)
  • {1,,1,,1,,1,,2} has recursion level C(C(Ω22+C(Ω2+d2+C(Ω22,0),0),0),0)
  • {1,,1,,1,,1{1,,1,,1,,1,,2}1,2} has recursion level C(C(Ω22+C(Ω2+d2+C(Ω22,0)+1,0),0),0)
  • {1,,2,,1,,1,,2} has recursion level C(C(Ω22+C(Ω2+d2+C(d,C(Ω22,0)),0),0),0)
  • {1,,1{1,,2,,1,,1,,2}2,,1,,1,,2} has recursion level C(C(Ω22+C(Ω2+d2+C(C(Ω2+d,C(Ω22,0)),C(Ω22,0)),0),0),0)
  • {1,,1{1,,2,,1,,1,,2}1{1,,2,,1,,1,,2}2,,1,,1,,2} has recursion level C(C(Ω22+C(Ω2+d2+C(C(Ω2+d2,C(Ω22,0)),C(Ω22,0)),0),0),0)
  • {1,,1{1,,2,,1,,1,,2}1{1,,2,,1,,1,,2}1{1,,2,,1,,1,,2}2,,1,,1,,2} has recursion level C(C(Ω22+C(Ω2+d2+C(C(Ω22,C(Ω22,0)),C(Ω22,0)),0),0),0)
  • {1,,1{1,,2,,1,,1,,2}1{1,,2,,1,,1,,2}1{1,,2,,1,,1,,2}1,2,,1,,1,,2} has recursion level C(C(Ω22+C(Ω2+d2+C(C(Ω22+1,0),C(Ω22,0)),0),0),0)
  • {1,,1{1,,2,,1,,1,,2}1{1,,2,,1,,1,,2}1{1,,2,,1,,1,,2}1{1,,1,,1,,1,,2}2,,1,,1,,2} has recursion level C(C(Ω22+C(Ω2+d3,0),0),0)
  • {1,,1{1,,1,,2,,1,,2}1,2,,1,,1,,2} has recursion level C(C(Ω22+C(Ω2+C(Ω22,0)+1,C(Ω2+d3,0)),0),0)
  • {1,,1,,1,2,,1,,2} has recursion level C(C(Ω22+C(Ω2+d+1,C(Ω2+d3,0)),0),0)
  • {1,,1,,1,,1,2,,2} has recursion level C(C(Ω22+C(Ω2+d2+1,C(Ω2+d3,0)),0),0)
  • {1,,1,,1,,1,,3} has recursion level C(C(Ω22+C(Ω2+d2+C(Ω22,0),C(Ω2+d3,0)),0),0)
  • {1,,1{1,,2,,1,,1,,3}1{1,,2,,1,,1,,3}1{1,,2,,1,,1,,3}1{1,,1,,1,,1,,3}2,,1,,1,,3} has recursion level C(C(Ω22+C(Ω2+d3,C(Ω2+d3,0)),0),0)
  • {1,,1,,1,,1,,1,2} has recursion level C(C(Ω22+C(Ω2+d3+1,0),0),0)
  • {1,,1,,1,,1,,1,,2} has recursion level C(C(Ω22+C(Ω2+d3+C(Ω22,0),0),0),0)
  • {1,,1,,1,,1,,1,,1,,2} has recursion level C(C(Ω22+C(Ω2+d4+C(Ω22,0),0),0),0)
  • {1{2,,}2} has recursion level C(C(Ω22+C(Ω2+ω^(d+1),0),0),0)

Something strange appears here. The recursion level of {1{1{2,,}2}2} is only equal to that of {1{1,,2}2}, because s(n,n{1{1{2,,}2}2}2) = s(n,n{1{1,,1,,…1,,1,,2}2}2) = (case B2, step 6) s(n,n{1{1…{1,2}…2}2}2) with n-1 1′s. Then {1{1{1{2,,}2}2{2,,}2}2} has equal recursion level to {1{1,,2}2,,2}. Also, {1,,1{1{1{2,,}2}2{2,,}2}2} has equal recursion level to {1{1,,1,,2}2,,1,,2}, {1,,1,,1{1{1{2,,}2}2{2,,}2}2} has equal recursion level to {1{1,,1,,1,,2}2,,1,,1,,2}, and so on – they’re all less than the recursion level of {1{2,,}2}.

However, {1{1,,2}2{2,,}2} has equal recursion level to {1{1,,2}2,,2}, {1{1,,1,,2}2{2,,}2} has equal recursion level to {1{1,,1,,2}2,,1,,2}, {1{1,,1,,1,,2}2{2,,}2} has equal recursion level to {1{1,,1,,1,,2}2,,1,,1,,2}, and so on, so {1{1{2,,}2}2{2,,}2} has equal recursion level to {1{2,,}2}.

What’s more, this “recursion level platform” happens on higher level of separators, such as {1{1{2,,}2}3{2,,}2}, {1{1{2,,}2}1{1{2,,}2}2{2,,}2}, {1{1{1{2,,}2}2{2,,}2}2{2,,}2}, {1,,2{2,,}2}, {1,,3{2,,}2}, {1,,1,,2{2,,}2}, {1,,1,,1,,2{2,,}2}, and so on – until {1{2,,}3}. Inside {1 ____ 2}, the {1{2,,}2} actually works in the same way as {1,,2}. Further, {1,,2{2,,}2} actually works as {1,,3}, {1,,1,,2{2,,}2} actually works as {1,,1,,2}, and {1{2,,}3} actually works as the real {1{2,,}2} – {1{1{2,,}3}2} has equal recursion level to {1{2,,}2}.

Also, {1,,1{1{2,,}3}2}, {1,,1,,1{1{2,,}3}2}, and even the single {1{2,,}3}, all have equal recursion level to {1{2,,}2}. The main reason of the “recursion level platform” described above is at the step 5 in case B2. For separator {1,,1,,…1,,1,,2} (which is resulted from {1{2,,}2}), the “X ,, n-1 Y” is {1,,1,,…1,,1,,1}, then lv(“X ,, n-1 Y”) = lv({1}), so we always turn into step 6 (expanding rule), with no adding between the separator {1,,1,,…1,,1,,2} and the outside {1 ____ 2}. However, for separator {1,,1,,…1,,1,,2{2,,}2} (which is resulted from {1{2,,}3}), the “X ,, n-1 Y” is {1,,1,,…1,,1,,1{2,,}2}, then lv(“X ,, n-1 Y”) = lv({1{2,,}2}), so for an outside layer lower-leveled than {1{2,,}2} (e.g. the {1 ____ 2} layer), we can turn into step 7 (adding rule), then we leave the “recursion level platform”.

“Recursion level platform” also happens on higher {1{2,,}1{2,,}2}, {1{3,,}2}, {1{1,2,,}2}, {1{1`2,,}2}, {1{1{1{1,2,,}2}2,,}2}, {1{1{1{1,,2,,}2}2,,}2} and even {1{1{1,,2{1,,2,,}2}2,,}2}. But {1{2,,2,,}2} reduces to {1{1,,2,,}…1{1,,2,,}1{1,,2,,}2}, and D({1{1,,2,,}…1{1,,2,,}1{1,,2,,}2}) = {1{1,,2,,}…1{1,,2,,}1{1,,1,,}2} = (rule 2) {1{1,,2,,}…1{1,,2,,}1,,2}, so there’s no “recursion level platform” here.

Now let ordinal d = C(Ω2,C(Ω22,0)).

  • {1,,1{1{2,,}3}2} has recursion level C(C(Ω22+C(Ω2+ω^(d+1),0),0),0)
  • {1,,1{1,,1{1{2,,}3}2}1{1,,2{1,,1,,2}2}2{1{2,,}3}2} has recursion level C(C(Ω22+C(Ω2+ω^(d+1),0)+C(Ω2,C(Ω2+C(Ω22,0),0)),0),0)
  • {1,,1{1,,1{1{2,,}3}2}1{1,,1{1{2,,}3}2}2{1{2,,}3}2} has recursion level C(C(Ω22+C(Ω2+ω^(d+1),0)+C(Ω2+C(C(Ω22+C(Ω2+ω^(d+1),0),0),C(Ω22,0)),0),0),0)
  • {1,,1{1{2,,}3}1,2} has recursion level C(C(Ω22+C(Ω2+ω^(d+1),0)+C(Ω2+C(C(Ω22+C(Ω2+ω^(d+1),0),0),C(Ω22,0))+1,0),0),0)
  • {1,,1{2{2,,}3}2} has recursion level C(C(Ω22+C(Ω2+ω^(d+1),0)+C(Ω2+ω^(C(C(Ω22+C(Ω2+ω^(d+1),0),0),C(Ω22,0))+1),0),0),0)
  • {1{1,,1,,2}2{2,,}3} has recursion level C(C(Ω22+C(Ω2+ω^(d+1),0)+C(Ω2+ω^(C(C(Ω22+C(Ω2+ω^(d+1),0),0),C(Ω22,0))+C(Ω22,0)),0),0),0)
  • {1,,1{1,,1{1,,1,,1{1{2,,}3}2}2,,2}1{1,,2,,2}2{1,,1,,1{1{2,,}3}2}2,,2} has recursion level C(C(Ω22+C(Ω2+ω^(d+1),0)+C(Ω2,C(Ω2+d,0)),0),0)
  • {1,,1{1,,1{1,,1,,1{1{2,,}3}2}2,,2}1{1,,1{1,,1,,1{1{2,,}3}2}2,,2}2{1,,1,,1{1{2,,}3}2}2,,2} has recursion level C(C(Ω22+C(Ω2+ω^(d+1),0)+C(Ω2+C(C(Ω22+C(Ω2+ω^(d+1),0),0),C(Ω22,0)),C(Ω2+d,0)),0),0)
  • {1,,1{1,,1,,1{1{2,,}3}2}1,2,,2} has recursion level C(C(Ω22+C(Ω2+ω^(d+1),0)+C(Ω2+C(C(Ω22+C(Ω2+ω^(d+1),0),0),C(Ω22,0))+1,C(Ω2+d,0)),0),0)
  • {1{1,,1,,3}2,,1,,1{1{2,,}3}2} has recursion level C(C(Ω22+C(Ω2+ω^(d+1),0)+C(Ω2+ω^(C(C(Ω22+C(Ω2+ω^(d+1),0),0),C(Ω22,0))+C(Ω22,0)),C(Ω2+d,0)),0),0)
  • {1{1,,1,,1,2}2,,1,,1{1{2,,}3}2} has recursion level C(C(Ω22+C(Ω2+ω^(d+1),0)+C(Ω2+d+1,0),0),0)
  • {1{1,,1,,1{1{2,,}2}2}2,,1,,1{1{2,,}3}2} has recursion level C(C(Ω22+C(Ω2+ω^(d+1),0)+C(Ω2+d+C(Ω22,0),0),0),0)
  • {1{1,,1,,1{1{2,,}3}2}2,,1,,1{1{2,,}3}2} has recursion level C(C(Ω22+C(Ω2+ω^(d+1),0)+C(Ω2+d+C(C(Ω22+C(Ω2+ω^(d+1),0),0),C(Ω22,0)),0),0),0)
  • {1,,1{1,,1,,2{1{2,,}3}2}1,2,,1{1{2,,}3}2} has recursion level C(C(Ω22+C(Ω2+ω^(d+1),0)+C(Ω2+C(Ω22,0)+1,C(Ω2+d+C(C(Ω22+C(Ω2+ω^(d+1),0),0),C(Ω22,0)),0)),0),0)
  • {1,,1{1,,2,,2{1{2,,}3}2}1{1,,1,,2{1{2,,}3}2}2,,2{1{2,,}3}2} has recursion level C(C(Ω22+C(Ω2+ω^(d+1),0)+C(Ω2+d,C(Ω2+d+C(C(Ω22+C(Ω2+ω^(d+1),0),0),C(Ω22,0)),0)),0),0)
  • {1,,1,,1,2{1{2,,}3}2} has recursion level C(C(Ω22+C(Ω2+ω^(d+1),0)+C(Ω2+d+1,C(Ω2+d+C(C(Ω22+C(Ω2+ω^(d+1),0),0),C(Ω22,0)),0)),0),0)
  • {1,,1,,1{1{2,,}3}1,2} has recursion level C(C(Ω22+C(Ω2+ω^(d+1),0)+C(Ω2+d+C(C(Ω22+C(Ω2+ω^(d+1),0),0),C(Ω22,0))+1,0),0),0)
  • {1,,1,,1{2{2,,}3}2} has recursion level C(C(Ω22+C(Ω2+ω^(d+1),0)+C(Ω2+d+ω^(C(C(Ω22+C(Ω2+ω^(d+1),0),0),C(Ω22,0))+1),0),0),0)
  • {1{1,,1,,1,,2}2{2,,}3} has recursion level C(C(Ω22+C(Ω2+ω^(d+1),0)+C(Ω2+d+ω^(C(C(Ω22+C(Ω2+ω^(d+1),0),0),C(Ω22,0))+C(Ω22,0)),0),0),0)
  • {1{1,,1,,1,,1,,2}2{2,,}3} has recursion level C(C(Ω22+C(Ω2+ω^(d+1),0)+C(Ω2+d2+ω^(C(C(Ω22+C(Ω2+ω^(d+1),0),0),C(Ω22,0))+C(Ω22,0)),0),0),0)
  • {1{1{2,,}2}2{2,,}3} has recursion level C(C(Ω22+C(Ω2+ω^(d+1),0)2,0),0)
  • {1,,1{1{1{2,,}3}2{2,,}3}2} has recursion level C(C(Ω22+C(Ω2+ω^(d+1),0)2,0),0)
  • {1{1{2,,}3}2{2,,}3} has recursion level C(C(Ω22+C(Ω2+ω^(d+1),0)2,0),0)
  • {1{1{2,,}3}1,2{2,,}3} has recursion level C(C(Ω22+ω^(C(Ω2+ω^(d+1),0)+1),0),0)
  • {1,,2{2,,}3} has recursion level C(C(Ω22+C(C(Ω2,C(Ω2+ω^(d+1),0)),C(Ω2+ω^(d+1),0)),0),0)
  • {1,,1,,2{2,,}3} has recursion level C(C(Ω22+C(C(Ω22,0),C(Ω2+ω^(d+1),0)),0),0)
  • {1,,1{1,,1,,2{2,,}3}1,2,,1{2,,}3} has recursion level C(C(Ω22+C(Ω2+C(Ω22,0)+1,C(Ω2+ω^(d+1),0)),0),0)
  • {1,,1{1,,2,,2{2,,}3}1{1,,1,,2{2,,}3}2,,2{2,,}3} has recursion level C(C(Ω22+C(Ω2+d,C(Ω2+ω^(d+1),0)),0),0)
  • {1,,1,,1,2{2,,}3} has recursion level C(C(Ω22+C(Ω2+d+1,C(Ω2+ω^(d+1),0)),0),0)
  • {1,,1,,1,,2{2,,}3} has recursion level C(C(Ω22+C(Ω2+d+C(Ω22,0),C(Ω2+ω^(d+1),0)),0),0)
  • {1,,1,,1,,1,,2{2,,}3} has recursion level C(C(Ω22+C(Ω2+d2+C(Ω22,0),C(Ω2+ω^(d+1),0)),0),0)
  • {1{2,,}5} has recursion level C(C(Ω22+C(Ω2+ω^(d+1),C(Ω2+ω^(d+1),0)),0),0)
  • {1{2,,}7} has recursion level C(C(Ω22+C(Ω2+ω^(d+1),C(Ω2+ω^(d+1),C(Ω2+ω^(d+1),0))),0),0)
  • {1{2,,}1,2} has recursion level C(C(Ω22+C(Ω2+ω^(d+1)+1,0),0),0)
  • {1{2,,}1,,2} has recursion level C(C(Ω22+C(Ω2+ω^(d+1)+C(Ω22,0),0),0),0)
  • {1{2,,}1{1{2,,}1,,2}1,2} has recursion level C(C(Ω22+C(Ω2+ω^(d+1)+C(Ω22,0)+1,0),0),0)
  • {1,,1{1{2,,}1,,2}2{2,,}1,,2} has recursion level C(C(Ω22+C(Ω2+ω^(d+1)+C(C(Ω2+C(Ω22,0),C(Ω22,0)),C(Ω22,0)),0),0),0)
  • {1,,1{1,,2{2,,}1,,2}2{2,,}1,,2} has recursion level C(C(Ω22+C(Ω2+ω^(d+1)+C(C(Ω2+d,C(Ω22,0)),C(Ω22,0)),0),0),0)
  • {1,,1{1,,2{2,,}1,,2}1,2{2,,}1,,2} has recursion level C(C(Ω22+C(Ω2+ω^(d+1)+C(C(Ω2+d+1,C(Ω22,0)),C(Ω22,0)),0),0),0)
  • {1,,1{1,,2{2,,}1,,2}1{1,,2{2,,}1,,2}2{2,,}1,,2} has recursion level C(C(Ω22+C(Ω2+ω^(d+1)+C(C(Ω2+d2,C(Ω22,0)),C(Ω22,0)),0),0),0)
  • {1,,1{2,,2{2,,}1,,2}2{2,,}1,,2} has recursion level C(C(Ω22+C(Ω2+ω^(d+1)+C(C(Ω2+ω^(d+1),C(Ω22,0)),C(Ω22,0)),0),0),0)
  • {1,,1{2,,2{2,,}1,,2}1,2{2,,}1,,2} has recursion level C(C(Ω22+C(Ω2+ω^(d+1)+C(C(Ω2+ω^(d+1)+1,C(Ω22,0)),C(Ω22,0)),0),0),0)
  • {1,,1{2,,2{2,,}1,,2}1{1,,2{2,,}1,,2}2{2,,}1,,2} has recursion level C(C(Ω22+C(Ω2+ω^(d+1)+C(C(Ω22,C(Ω22,0)),C(Ω22,0)),0),0),0)
  • {1,,1{2,,2{2,,}1,,2}1{1,,2{2,,}1,,2}1,2{2,,}1,,2} has recursion level C(C(Ω22+C(Ω2+ω^(d+1)+C(C(Ω22+1,0),C(Ω22,0)),0),0),0)
  • {1,,1{2,,2{2,,}1,,2}1{1,,2{2,,}1,,2}1{1{2,,}1,,2}2{2,,}1,,2} has recursion level C(C(Ω22+C(Ω2+ω^(d+1)+d,0),0),0)
  • {1{2,,}1,,1,2} has recursion level C(C(Ω22+C(Ω2+ω^(d+1)+d+1,0),0),0)
  • {1{2,,}1,,1,,2} has recursion level C(C(Ω22+C(Ω2+ω^(d+1)+d+C(Ω22,0),0),0),0)
  • {1{2,,}1,,1,,1,,2} has recursion level C(C(Ω22+C(Ω2+ω^(d+1)+d2+C(Ω22,0),0),0),0)
  • {1{2,,}1{2,,}2} has recursion level C(C(Ω22+C(Ω2+ω^(d+1)2,0),0),0)
  • {1{3,,}2} has recursion level C(C(Ω22+C(Ω2+ω^(d+2),0),0),0)
  • {1{1{1{2,,}2}2,,}2} has recursion level C(C(Ω22+C(Ω2+ω^(d+C(Ω2+ω^(d+1),0)),0),0),0)
  • {1{1,,2,,}2} has recursion level C(C(Ω22+C(Ω2+ω^(d+C(Ω22,0)),0),0),0)

Now let separator ■ = {1{1,,2,,}2} and let ordinal d = C(Ω2,C(Ω22,0)), and ordinal a = C(Ω2+ω^(d+C(Ω22,0)),0) = C(Ω2+ω^(C(Ω2,C(Ω22,0))+C(Ω22,0)),0).

  • {1{1■2,,}3} has recursion level C(C(Ω22+a,0),0)
  • {1,,1{1,,1{1{1■2,,}3}2}1{1,,1{1{1■2,,}3}2}2{1{1■2,,}3}2} has recursion level C(C(Ω22+a+C(Ω2+C(C(Ω22+a,0),C(Ω22,0)),0),0),0)
  • {1,,1{1{1■2,,}3}1,2} has recursion level C(C(Ω22+a+C(Ω2+C(C(Ω22+a,0),C(Ω22,0))+1,0),0),0)
  • {1{1,,1,,2}2{1■2,,}3} has recursion level C(C(Ω22+a+C(Ω2+ω^(C(C(Ω22+a,0),C(Ω22,0))+C(Ω22,0)),0),0),0)
  • {1{1,,1,,1,,2}2{1■2,,}3} has recursion level C(C(Ω22+a+C(Ω2+d+ω^(C(C(Ω22+a,0),C(Ω22,0))+C(Ω22,0)),0),0),0)
  • {1{1,,1,,1,,1,,2}2{1■2,,}3} has recursion level C(C(Ω22+a+C(Ω2+d2+ω^(C(C(Ω22+a,0),C(Ω22,0))+C(Ω22,0)),0),0),0)
  • {1{1{2,,}2}2{1■2,,}3} has recursion level C(C(Ω22+a+C(Ω2+ω^(d+1),0),0),0)
  • {1{1{1■2,,}3}2{1■2,,}3} has recursion level C(C(Ω22+a2,0),0)
  • {1,,2{1■2,,}3} has recursion level C(C(Ω22+C(C(Ω2,a),a),0),0)
  • {1,,1,,2{1■2,,}3} has recursion level C(C(Ω22+C(C(Ω22,0),a),0),0)
  • {1,,1,,1,,2{1■2,,}3} has recursion level C(C(Ω22+C(Ω2+d+C(Ω22,0),a),0),0)
  • {1,,1,,1,,1,,2{1■2,,}3} has recursion level C(C(Ω22+C(Ω2+d2+C(Ω22,0),a),0),0)
  • {1{2,,}2{1■2,,}3} has recursion level C(C(Ω22+C(Ω2+ω^(d+1),a),0),0)
  • {1{1{1{1■2,,}3}2,,}2{1■2,,}3} has recursion level C(C(Ω22+C(Ω2+ω^(d+a),a),0),0)
  • {1{1{1{1■2,,}3}2,,}1{2,,}2{1■2,,}3} has recursion level C(C(Ω22+C(Ω2+ω^(d+a)+ω^(d+1),a),0),0)
  • {1{2{1{1■2,,}3}2,,}2{1■2,,}3} has recursion level C(C(Ω22+C(Ω2+ω^(d+a+1),a),0),0)
  • {1{1{1{1,,2{1■2,,}3}2{1■2,,}3}2,,}2{1■2,,}3} has recursion level C(C(Ω22+C(Ω2+ω^(d+C(C(Ω2,a),a)),a),0),0)
  • {1{1{1{1{1{1{1■2,,}3}2,,}2{1■2,,}3}2{1■2,,}3}2,,}2{1■2,,}3} has recursion level C(C(Ω22+C(Ω2+ω^(d+C(C(Ω22+C(Ω2+ω^(d+a),a),0),a)),a),0),0)
  • {1{1{1,,2{1■2,,}3}2,,}2{1■2,,}3} has recursion level C(C(Ω22+C(Ω2+ω^(d+C(Ω2,a)),a),0),0)
  • {1{1{1{2,,}2{1■2,,}3}2,,}2{1■2,,}3} has recursion level C(C(Ω22+C(Ω2+ω^(d+C(Ω2+ω^(d+1),a)),a),0),0)
  • {1{1{1{1{1{1■2,,}3}2,,}2{1■2,,}3}2,,}2{1■2,,}3} has recursion level C(C(Ω22+C(Ω2+ω^(d+C(Ω2+ω^(d+a),a)),a),0),0)
  • {1{1■2,,}5} has recursion level C(C(Ω22+C(Ω2+ω^(d+C(Ω22,0)),a),0),0)
  • {1{1■2,,}1,2} has recursion level C(C(Ω22+C(Ω2+ω^(d+C(Ω22,0))+1,0),0),0)
  • {1{1■2,,}1,,2} has recursion level C(C(Ω22+C(Ω2+ω^(d+C(Ω22,0))+C(Ω22,0),0),0),0)
  • {1{1■2,,}1{2,,}2} has recursion level C(C(Ω22+C(Ω2+ω^(d+C(Ω22,0))+ω^(d+1),0),0),0)
  • {1{2■2,,}2} has recursion level C(C(Ω22+C(Ω2+ω^(d+C(Ω22,0)+1),0),0),0)
  • {1{1■3,,}2} has recursion level C(C(Ω22+C(Ω2+ω^(d+C(Ω22,0)2),0),0),0)
  • {1{1■1,2,,}2} has recursion level C(C(Ω22+C(Ω2+ω^(d+ω^(C(Ω22,0)+1)),0),0),0)
  • {1{1{2{1,,2,,}2}2,,}2} has recursion level C(C(Ω22+C(Ω2+ω^(d+ω^ω^(C(Ω22,0)+1)),0),0),0)
  • {1,,2{1,,2,,}2} has recursion level C(C(Ω22+C(Ω2+ω^(d+C(d,C(Ω22,0))),0),0),0)
  • {1,,1{1,,2{1,,2,,}2}2{1,,2,,}2} has recursion level C(C(Ω22+C(Ω2+ω^(d+C(C(Ω2+d,C(Ω22,0)),C(Ω22,0))),0),0),0)
  • {1,,1{2,,2{1,,2,,}2}2{1,,2,,}2} has recursion level C(C(Ω22+C(Ω2+ω^(d+C(C(Ω2+ω^(d+1),C(Ω22,0)),C(Ω22,0))),0),0),0)
  • {1,,1{1■2,,2{1,,2,,}2}2{1,,2,,}2} has recursion level C(C(Ω22+C(Ω2+ω^(d+C(C(Ω2+ω^(d+C(Ω22,0)),C(Ω22,0)),C(Ω22,0))),0),0),0)
  • {1,,1{1{1,,2{1,,2,,}2}2,,2{1,,2,,}2}2{1,,2,,}2} has recursion level C(C(Ω22+C(Ω2+ω^(d+C(C(Ω22,C(Ω22,0)),C(Ω22,0))),0),0),0)
  • {1,,1{1{1,,2{1,,2,,}2}2,,2{1,,2,,}2}1,2{1,,2,,}2} has recursion level C(C(Ω22+C(Ω2+ω^(d+C(C(Ω22+1,0),C(Ω22,0))),0),0),0)
  • {1,,1{1{1,,2{1,,2,,}2}2,,2{1,,2,,}2}1{1{1■2,,}2}2{1,,2,,}2} has recursion level C(C(Ω22+C(Ω2+ω^(d+C(C(Ω22+a,0),C(Ω22,0))),0),0),0)
  • {1,,1{1{1,,2{1,,2,,}2}2,,2{1,,2,,}2}1■2{1,,2,,}2} has recursion level C(C(Ω22+C(Ω2+ω^(d2),0),0),0)
  • {1,,1{1{1,,2{1,,2,,}2}2,,2{1,,2,,}2}1■1■2{1,,2,,}2} has recursion level C(C(Ω22+C(Ω2+ω^(d2),0)2,0),0)
  • {1,,1,,2{1,,2,,}2} has recursion level C(C(Ω22+C(C(Ω22,0),C(Ω2+ω^(d2),0)),0),0)
  • {1,,1,,1,,2{1,,2,,}2} has recursion level C(C(Ω22+C(Ω2+d+C(Ω22,0),C(Ω2+ω^(d2),0)),0),0)
  • {1,,1,,1,,1,,2{1,,2,,}2} has recursion level C(C(Ω22+C(Ω2+d2+C(Ω22,0),C(Ω2+ω^(d2),0)),0),0)
  • {1{2,,}2{1,,2,,}2} has recursion level C(C(Ω22+C(Ω2+ω^(d+1),C(Ω2+ω^(d2),0)),0),0)
  • {1{1■2,,}2{1,,2,,}2} has recursion level C(C(Ω22+C(Ω2+ω^(d+C(Ω2+ω^(d2),0)),C(Ω2+ω^(d2),0)),0),0)
  • {1{1,,2,,}3} has recursion level C(C(Ω22+C(Ω2+ω^(d+C(Ω22,0)),C(Ω2+ω^(d2),0)),0),0)
  • {1{1,,2,,}4} has recursion level C(C(Ω22+C(Ω2+ω^(d+C(Ω22,0)),C(Ω2+ω^(d2),C(Ω2+ω^(d2),0))),0),0)
  • {1{1,,2,,}1,2} has recursion level C(C(Ω22+C(Ω2+ω^(d2)+1,0),0),0)

Now let ordinal d = C(Ω2,C(Ω22,0)).

  • {1{1,,2,,}1{2,,}2} has recursion level C(C(Ω22+C(Ω2+ω^(d2)+ω^(d+1),0),0),0)
  • {1{1,,2,,}1{1,,2,,}2} has recursion level C(C(Ω22+C(Ω2+ω^(d2)+ω^(d+C(Ω22,0)),0),0),0)
  • {1{1,,2,,}1{1,,2,,}1{1,,2,,}2} has recursion level C(C(Ω22+C(Ω2+ω^(d2)2+ω^(d+C(Ω22,0)),0),0),0)
  • {1{2,,2,,}2} has recursion level C(C(Ω22+C(Ω2+ω^(d2+1),0),0),0)
  • {1{1,,3,,}2} has recursion level C(C(Ω22+C(Ω2+ω^(d2+C(Ω22,0)),0),0),0)
  • {1,,1,,2{1,,3,,}2} has recursion level C(C(Ω22+C(C(Ω22,0),C(Ω2+ω^(d3),0)),0),0)
  • {1{1,,3,,}1,2} has recursion level C(C(Ω22+C(Ω2+ω^(d3)+1,0),0),0)
  • {1{2,,3,,}2} has recursion level C(C(Ω22+C(Ω2+ω^(d3+1),0),0),0)
  • {1{1,,4,,}2} has recursion level C(C(Ω22+C(Ω2+ω^(d3+C(Ω22,0)),0),0),0)
  • {1{1,,1,2,,}2} has recursion level C(C(Ω22+C(Ω2+ω^ω^(d+1),0),0),0)
  • {1{1,,1,,2,,}2} has recursion level C(C(Ω22+C(Ω2+ω^ω^(d+C(Ω22,0)),0),0),0)
  • {1,,1,,2{1,,1,,2,,}2} has recursion level C(C(Ω22+C(C(Ω22,0),C(Ω2+ω^ω^(d2),0)),0),0)
  • {1{1,,1,,2,,}1,2} has recursion level C(C(Ω22+C(Ω2+ω^ω^(d2)+1,0),0),0)
  • {1{2,,1,,2,,}2} has recursion level C(C(Ω22+C(Ω2+ω^(ω^(d2)+1),0),0),0)
  • {1{1,,1,,3,,}2} has recursion level C(C(Ω22+C(Ω2+ω^(ω^(d2)+ω^(d+C(Ω22,0))),0),0),0)
  • {1{1,,1,,1,2,,}2} has recursion level C(C(Ω22+C(Ω2+ω^ω^(d2+1),0),0),0)
  • {1{1,,1,,1,,2,,}2} has recursion level C(C(Ω22+C(Ω2+ω^ω^(d2+C(Ω22,0)),0),0),0)
  • {1{1{2,,}2,,}2} has recursion level C(C(Ω22+C(Ω2+ω^ω^ω^(d+1),0),0),0)
  • {1{1{1,,2,,}2,,}2} has recursion level C(C(Ω22+C(Ω2+ω^ω^ω^(d+C(Ω22,0)),0),0),0)
  • {1{1{1,,1,,2,,}2,,}2} has recursion level C(C(Ω22+C(Ω2+ω^ω^ω^ω^(d+C(Ω22,0)),0),0),0)
  • {1{1{1{1,,2,,}2,,}2,,}2} has recursion level C(C(Ω22+C(Ω2+ω^ω^ω^ω^ω^(d+C(Ω22,0)),0),0),0)

Let f(n) = s(n,n{1{1{1…{1,2,,}…2,,}2,,}2}2) where there’re n separators with a double comma. So f(n) is the limit of pDAN. It eventually outgrows all above, and has growth rate C(C(Ω22+C(Ω2+C(C(Ω2,C(Ω2,C(Ω22,0))),C(Ω2,C(Ω22,0))),0),0),0) = C(C(Ω22+C(Ω2C(Ω2,C(Ω22,0))+1,0),0),0).

Numbers from extended array notation

Here are some typical numbers from extended array notation. They’re also examples to help you understand exAN.

Update note: those numbers are defined using an older version of exAN without the Rule 4.

Two row series

Those numbers are defined using expression s(a1,a2,…ak-1,ak{2}b1,b2,…bk-1,bk). The simplest ones are already defined in LAN numbers page, e.g.

In s(3,2{2}2), we start the process. We land at the 3rd entry – the 2 in s(3,2{2}2), and we meet case B2, so change s(3,2{2}2) into s(3,2{2}2{2}1). Then we focus on the first entry of the former {2}, and we meet case B3, so change s(3,2{2}2{2}1) into s(3,2{1}1{1}2{2}1). Since {1} is the comma, we get s(3,2{2}2) = s(3,2,1,2{2}1). That’s how exAN rules work. As a result,

s(3,2{2}2) = s(3,2,1,2{2}1) = s(3,2,1,2) = s(3,3,2,1) = s(3,3,2) = tribo
s(3,3{2}2) = s(3,3,1,1,2{2}1) = s(3,3,1,1,2) = s(3,3,3,3,1) = s(3,3,3,3) = tetentri
s(3,4{2}2) = s(3,4,1,1,1,2{2}1) = s(3,4,1,1,1,2) = s(3,3,3,3,4,1) = s(3,3,3,3,4) = truchaindupribolplex
s(4,2{2}2) = s(4,2,1,2{2}1) = s(4,2,1,2) = s(4,4,2,1) = s(4,4,2) = tetbo
s(4,4{2}2) = s(4,4,1,1,1,2{2}1) = s(4,4,1,1,1,2) = s(4,4,4,4,4,1) = s(4,4,4,4,4) = pententet
s(5,2{2}2) = s(5,2,1,2{2}1) = s(5,2,1,2) = s(5,5,2,1) = s(5,5,2) = pentbo
s(5,5{2}2) = s(5,5,1,1,1,1,2{2}1) = s(5,5,1,1,1,1,2) = s(5,5,5,5,5,5,1) = s(5,5,5,5,5,5) = hexenpent
s(6,2{2}2) = s(6,2,1,2{2}1) = s(6,2,1,2) = s(6,6,2,1) = s(6,6,2) = hexabo

Some larger numbers are listed below.

Linel group

This group includes linel, linelplex, lintrienel, lintetenel and linpentenel in order.

Linel = s(3,2,2{2}2) = s(3,s(3,1,2{2}2),1{2}2) = s(3,s(3,1,1{2}2),1{2}2) = s(3,s(3,1,1,2{2}1),1{2}2) = s(3,s(3,1,1,2),1{2}2) = s(3,s(3,3,1,1),1{2}2) = s(3,s(3,3),1{2}2) = s(3,27,1{2}2) = s(3,27,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2{2}1) = s(3,27,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2) = s(3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,27,1) = s(3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,27) (with 28 3′s).

Linelplex = s(3,3,2{2}2) = s(3,s(3,2,2{2}2),1{2}2) = s(3,s(3,2,2{2}2),1,1…,1,2{2}1) = s(3,s(3,2,2{2}2),1,1…,1,2) (with s(3,2,2{2}2) 1′s) = s(3,3,…3,3,s(3,2,2{2}2),1) = s(3,3,…3,3,s(3,2,2{2}2)) (with s(3,2,2{2}2) + 1 3′s). It’s so hard to express in LAN! Note: though s(3,2{2}2) = s(3,3,2), s(3,2{2}3) ≠ s(3,3,2{2}2). In fact, s(3,2{2}3) = s(3,2,1,2{2}2) = s(3,3,2,1{2}2) = s(3,3,…3,3,s(3,2,2{2}2)) (with s(3,2,2{2}2) + 2 3′s) – that’s because rule 3 only applies to “the 1 in the last entry”.

Lintrienel = s(3,3,3{2}2) = s(3,s(3,2,3{2}2),2{2}2) = s(3,s(3,s(3,1,3{2}2),2{2}2),2{2}2) = s(3,s(3,s(3,1,1{2}2),2{2}2),2{2}2) = s(3,s(3,27,2{2}2),2{2}2). The name “lintrienel” bases on “linel” and adds an “-trien-“, which means an “3,3,3” string in the “first row”.

Lintetenel = s(3,3,3,3{2}2) and linpentenel = s(3,3,3,3,3{2}2). And, for the same reason, s(3,3{2}3) ≠ s(3,3,3,3{2}2).

Dulinel group

This group includes dulinel, dulinelplex, dulintrienel and dulintetenel in order.

Dulinel = s(3,2,2{2}3) = s(3,s(3,1,2{2}3),1{2}3) = s(3,s(3,1,1{2}3),1{2}3) = s(3,s(3,1,1,2{2}2),1{2}3) = s(3,s(3,3,1,1{2}2),1{2}3) = s(3,s(3,3,1,1,1,1,2{2}1),1{2}3) = s(3,s(3,3,1,1,1,1,2),1{2}3) = s(3,hexentri,1{2}3) = s(3,hexentri,1,1…,1,2{2}2) (with hexentri 1′s) = s(3,3,…3,3,hexentri,1{2}2) (with hexentri + 1 3′s).

Dulinelplex = s(3,3,2{2}3) = s(3,s(3,2,2{2}3),1{2}3) = s(3,s(3,2,2{2}3),1{2}3) = s(3,dulinel,1{2}3) = s(3,dulinel,1,1…,1,2{2}2) (with dulinel 1′s) = s(3,3,…3,3,hexentri,1{2}2) (with dulinel + 1 3′s).

Dulintrienel = s(3,3,3{2}3) and dulintetenel = s(3,3,3,3{2}3).

Dienlinel group

This group includes dienlinel, dienlintrienel, trienlinel, tetenlinel and pentenlinel in order.

Dienlinel = s(3,2,2{2}3,3), dienlintrienel = s(3,3,3{2}3,3), trienlinel = s(3,2,2{2}3,3,3), tetenlinel = s(3,2,2{2}3,3,3,3) and pentenlinel = s(3,2,2{2}3,3,3,3,3). Here the “dien-“, “trien-“, “teten-” and “penten-” mean 2, 3, 4 or 5 entries of 3 respectively.

Dimensional series

Those numbers are defined using arrays with separator {n} only (where n is a positive integer). A special result is that s(a,1,1…,1{n}2 #) = s(a,1,1…,1{n-1}2 #), so s(a,1,1…,1{n}2 #) = s(a,1,1…,1,2 #).

Linbel group

This group includes linbel, lintrienbel, linblinel, linbdulinel, linbdienlinel, dulinbel, dienlinbel and trienlinbel in order.

Linbel = s(3,2,2{2}1{2}2) = s(3,s(3,1,2{2}1{2}2),1{2}1{2}2) = s(3,s(3,1,1{2}1{2}2),1{2}1{2}2) = s(3,s(3,1,1{2}1,2),1{2}1{2}2) = s(3,s(3,3,3{2}1,1),1{2}1{2}2) = s(3,trientri,1{2}1{2}2) = s(3,trientri,1{2}1,1,…1,1,2) with trientri 1′s. It marks the end of 2-row arrays (arrays with only one {2} and no separator higher than {2})!

Lintrienbel = s(3,3,3{2}1{2}2), linblinel = s(3,2,2{2}2{2}2), linbdulinel = s(3,2,2{2}3{2}2), linbdienlinel = s(3,2,2{2}3,3{2}2), dulinbel = s(3,2,2{2}1{2}3), dienlinbel = s(3,2,2{2}1{2}3,3) and trienlinbel = s(3,2,2{2}1{2}3,3,3).

Lintrel group

This group includes lintrel, lintetel, linpenel and linhexel in order.

Lintrel = s(3,2,2{2}1{2}1{2}2), lintetel = s(3,2,2{2}1{2}1{2}1{2}2), linpenel = s(3,2,2{2}1{2}1{2}1{2}1{2}2) and linhexel = s(3,2,2{2}1{2}1{2}1{2}1{2}1{2}2). They mark the end of 3-row, 4-row, 5-row and 6-row arrays respectively.

Planel group

This group includes planel, planelplex, plantrienel, planlinel and planlinbel in order.

Planel = s(3,2,2{3}2) = s(3,s(3,1,2{3}2),1{3}2) = s(3,s(3,1,1{3}2),1{3}2) = s(3,s(3,1,1{2}2),1{3}2) = s(3,s(3,1,1,2),1{3}2) = s(3,s(3,3,1,1),1{3}2) = s(3,27,1{3}2) = s(3,27,1{2}1{2}1{2}1{2}1{2}1{2}1{2}1{2}1{2}1{2}1{2}1{2}1{2}1{2}1{2}1{2}1{2}1{2}1{2}1{2}1{2}1{2}1{2}1{2}1{2}1{2}1{2}2). And planelplex = s(3,3,2{3}2) = s(3,planel,1{2}1{2}…1{2}1{2}2) with planel {2}′s – that marks the end of planar arrays (arrays with commas and {2}′s only)!

Plantrienel = s(3,3,3{3}2), planlinel = s(3,2,2{2}2{3}2) and planlinbel = s(3,2,2{2}1{2}2{3}2).

planbel group

This group includes duplanel, dienplanel, trienplanel, planbel and plantrel in order.

Duplanel = s(3,2,2{3}3), dienplanel = s(3,2,2{3}3,3), trienplanel = s(3,2,2{3}3,3,3), planbel = s(3,2,2{3}1{3}2) and plantrel = s(3,2,2{3}1{3}1{3}2).

Cubel group

This group includes cubel, tesseral and penteral in order.

Cubel = s(3,2,2{4}2) = s(3,s(3,1,2{4}2),1{4}2) = s(3,s(3,1,1{4}2),1{4}2) = s(3,s(3,1,1,2),1{4}2) = s(3,s(3,3,1,1),1{4}2) = s(3,27,1{4}2) = s(3,27,1{3}1{3}1{3}1{3}1{3}1{3}1{3}1{3}1{3}1{3}1{3}1{3}1{3}1{3}1{3}1{3}1{3}1{3}1{3}1{3}1{3}1{3}1{3}1{3}1{3}1{3}1{3}2).

Tesseral = s(3,2,2{5}2) = s(3,27,1{5}2) = s(3,27,1{4}1{4}1{4}1{4}1{4}1{4}1{4}1{4}1{4}1{4}1{4}1{4}1{4}1{4}1{4}1{4}1{4}1{4}1{4}1{4}1{4}1{4}1{4}1{4}1{4}1{4}1{4}2) – now you see how much tesseral is larger than cubel, but we only change the {4} into the {5}.

Finally penteral = s(3,2,2{6}2).

Post-dimensional series

In definition of those numbers, there’re at least layer-2 entries. Though rules for them are the same as before, the reducing process seems more complex. So let me explain more to you.

Dimentril group

This group includes dimensol, dimentriensol, dimenlinsol, dimenlinbsol, dimentril, diendimensol, lindimensol, and dimenbol in order.

Dimentril = s(3,3{1,2}3). To solve it, we start the process, and meet the 3 in s(3,3{1,2}3). It’s case B2, so change the “{1,2}3” into “{1,2}2{1,2}2“, and jump to the 1 in s(3,3{1,2}2{1,2}2). Then we meet the 2 in s(3,3{1,2}2{1,2}2). It’s case B1, so s(3,3{1,2}3) = s(3,3{3,1}2{1,2}2) = s(3,3{3}2{1,2}2). The next time we start the process, we first meet case B2 then case B3, and s(3,3{3}2{1,2}2) = s(3,3{2}1{2}1{2}2{1,2}2) = s(3,3{2}1{2}1,1,1,2{1,2}2) = s(3,3{2}3{2}3,3,3,1{1,2}2). Also s(3,1{1,2}1,1,2) = s(3,3{1,2}3,1,1) = dimentril, and s(3,3{1,2}1,2) = s(3,3{1,2}3,1) = dimentril. The name “dimentril” combines “dimension” and “-tril”, because it’s above dimensional arrays, and all entries at base layer are 3.

However, dimentril is too far above dimensional arrays, so let’s take some more steps to see its hugeness.

Dimensol = s(3,3{1,2}2) = s(3,3{3,1}2{1,2}1) = s(3,3{3}2) = s(3,3{2}1{2}1{2}2) = s(3,3{2}1{2}1,1,1,2) = s(3,3{2}3{2}3,3,3) – it’s not so large, even smaller than lintrel.

But dimentriensol is larger than all dimensional numbers. It’s s(3,3,3{1,2}2) = s(3,s(3,2,3{1,2}2),2{1,2}2) = s(3,s(3,s(3,1,3{1,2}2),2{1,2}2),2{1,2}2) = s(3,s(3,s(3,1,1{1,2}2),2{1,2}2),2{1,2}2) = s(3,s(3,s(3,3,3,2),2{1,2}2),2{1,2}2), where s(3,n,2{1,2}2) = s(3,s(3,n-1,2{1,2}2),1{1,2}2) = s(3,3,3{s(3,n-1,2{1,2}2)}2) – an s(3,n-1,2{1,2}2)-1 dimension array!

Dimenlinsol = s(3,3{2}2{1,2}2) = s(3,3,1,1,2{1,2}2) = s(3,3,3,3,1{1,2}2), and dimenlinbsol = s(3,3{2}1{2}2{1,2}2) = s(3,3{2}1,1,1,2{1,2}2) = s(3,3{2}3,3,3,1{1,2}2). And dimentril is the 3rd term beginning with dimenlinsol and dimenlinbsol.

Going beyond dimentril, diendimensol = s(3,3{1,2}3,3) = s(3,3{3}2{1,2}2,3), lindimensol = s(3,3{1,2}1{2}2) = s(3,3{1,2}1,1,1,2) = s(3,3{1,2}3,3,3), and dimenbol = s(3,3{1,2}1{1,2}2) = s(3,3{1,2}3{3,1}2{1,2}1) = s(3,3{1,2}3{3}2).

Dimensolplex group (New Naming System)

From now on, we start another naming system. In the system, we use 4 “namebases” to name numbers result from some typical expressions. They’re

  • “ol” – to name expressions s(3,3A2)
  • “trien” – to name expressions s(3,3,3A2)
  • “tril” – to name expressions s(3,3A3)
  • “bol” – to name expressions s(3,3A1A2)

where A is a separator. The first group beyond dimenbol is dimensolplex group. This group contains 19 numbers as follows. Though I use the same suffix “-plex” as before, they have fully different meaning.

  • dimensolplex = s(3,3{2,2}2) = s(3,3{1,2}1{1,2}1{1,2}2) = s(3,3{1,2}3{1,2}3{3}2). It’s similar to dimenbol, but with 3 “dimension rows” instead of 2.
  • dimentrienplex = s(3,3,3{2,2}2). Notice the change of the namebase from dimentril group. This time we have not just 3 dimension rows, because the {2,2} reduces only when you meet it in the process. Actually dimentrienplex = s(3,s(3,s(3,3,3{1,2}2),2{2,2}2),2{2,2}2), where s(3,n,2{2,2}2) reduces to an array with s(3,n-1,2{2,2}2) dimension rows!
  • dimentrilplex = s(3,3{2,2}3) = s(3,3{1,2}1{1,2}1{1,2}2{2,2}2) = s(3,3{1,2}3{1,2}3{3}2{2,2}2).
  • dimenbolplex = s(3,3{2,2}1{2,2}2) = s(3,3{2,2}1{1,2}1{1,2}1{1,2}2) = s(3,3{2,2}3{1,2}3{1,2}3{3}2).
  • dimentrienbiplex = s(3,3,3{3,2}2)
  • dimentrilbiplex = s(3,3{3,2}3)
  • dimenbolbiplex = s(3,3{3,2}1{3,2}2)
  • dimensoltriplex = s(3,3{4,2}2)
  • dimentrientriplex = s(3,3,3{4,2}2)
  • dimentriltriplex = s(3,3{4,2}3)
  • dimenboltriplex = s(3,3{4,2}1{4,2}2)
  • dimensolquadriplex = s(3,3{5,2}2)
  • dimentrienquadriplex = s(3,3,3{5,2}2)
  • dimentrilquadriplex = s(3,3{5,2}3)
  • dimenbolquadriplex = s(3,3{5,2}1{5,2}2)
  • dimensolquintiplex = s(3,3{6,2}2)
  • dimentrienquintiplex = s(3,3,3{6,2}2)
  • dimentrilquintiplex = s(3,3{6,2}3)
  • dimenbolquintiplex = s(3,3{6,2}1{6,2}2)

So the suffix n-plex means that the separator A becomes {n+1,2}. However, there’s one missing number…

-dex group

Now we pick up the missing number – dimensoldex, but you may think that the missing number is dimensolbiplex. Actually they’re the same thing, because

Dimensoldex = s(3,3{1,3}2) = s(3,3{3,2}2) = dimensolbiplex. The reducing process is similar to dimensol or dimentril. First we start the process, and meet the “2” in s(3,3{1,3}2) – that’s case B2, so change the “{1,3}2” into “{1,3}2{1,3}1″, and jump to the “1“. Then we meet the “3” in s(3,3{1,3}2{1,3}1) – that’s case B3, so s(3,3{1,3}2) = s(3,3{3,2}2{1,3}1) = s(3,3{3,2}2).

The suffix “-dex” change the {1,2} into {1,3}, so we get dimentriendex = s(3,3,3{1,3}2), dimentrildex = s(3,3{1,3}3) and dimenboldex = s(3,3{1,3}1{1,3}2).

The {1,3} is not just {3,2} – it’s {b,2} where b is the iterator only when you meet it in the process. Dimensoldex = dimensolbiplex – that’s a bit weak. But could you imagine what’s the n such that dimentriendex ≈ dimentrien-n-plex? Try to reduce it, and you’ll get similar answer to what I said about dimentriensol and dimentrienplex.

Then we can add “n-plex” suffixes to those numbers, such as

  • dimensoldexplex = s(3,3{2,3}2)
  • dimentriendexplex = s(3,3,3{2,3}2)
  • dimentrildexplex = s(3,3{2,3}3)
  • dimenboldexplex = s(3,3{2,3}1{2,3}2)
  • dimentriendexbiplex = s(3,3,3{3,3}2)
  • dimentrildexbiplex = s(3,3{3,3}3)
  • dimenboldexbiplex = s(3,3{3,3}1{3,3}2)

(still with one missing stuff…) So the suffix “n-plex” means the first entry of the separator A changed from 1 to n+1.

The next step is adding “-plex” prior to the “-dex”. And we get some larger numbers –

  • dimensolplexdex = s(3,3{1,4}2) = s(3,3{3,3}2), a.k.a. the missing stuff – dimensoldexbiplex.
  • dimentrienplexdex = s(3,3,3{1,4}2)
  • dimentrilplexdex = s(3,3{1,4}3)
  • dimenbolplexdex = s(3,3{1,4}1{1,4}2)
  • dimensolplexdexplex = s(3,3{2,4}2)
  • dimentrienplexdexplex = s(3,3,3{2,4}2)
  • dimentrilplexdexplex = s(3,3{2,4}3)
  • dimenbolplexdexplex = s(3,3{2,4}1{2,4}2)

And with “-biplex” prior to the “-dex”:

  • dimensolbiplexdex = s(3,3{1,5}2) = s(3,3{3,4}2)
  • dimentrienbiplexdex = s(3,3,3{1,5}2)
  • dimentrilbiplexdex = s(3,3{1,5}3)
  • dimenbolbiplexdex = s(3,3{1,5}1{1,5}2)
  • dimensolbiplexdexplex = s(3,3{2,5}2)
  • dimentrienbiplexdexplex = s(3,3,3{2,5}2)
  • dimentrilbiplexdexplex = s(3,3{2,5}3)
  • dimenbolbiplexdexplex = s(3,3{2,5}1{2,5}2)

With “-triplex” prior to the “-dex”:

  • dimensoltriplexdex = s(3,3{1,6}2) = s(3,3{3,5}2)
  • dimentrientriplexdex = s(3,3,3{1,6}2)
  • dimentriltriplexdex = s(3,3{1,6}3)
  • dimenboltriplexdex = s(3,3{1,6}1{1,6}2)

And so on. The “n-plex” prior to the “-dex” means adding n to the 2nd entry of the separator A.

-bidex group

Here’s how the suffix “-bidex” works.

Dimensolbidex = s(3,3{1,1,2}2). To solve it, first we start the process, and meet the “2” in s(3,3{1,1,2}2) – that’s case B2, so change the “{1,1,2}2” into “{1,1,2}2{1,1,2}1″, and jump to the “1“. Then we meet the “2” in s(3,3{1,1,2}2{1,1,2}1) – that’s case B3, so s(3,3{1,1,2}2) = s(3,3{1,3,1}2{1,1,2}1) = s(3,3{1,3}2) – it’s still dimensoldex (or dimensolbiplex).

  • dimentrienbidex = s(3,3,3{1,1,2}2) – that’s much larger.
  • dimentrilbidex = s(3,3{1,1,2}3)
  • dimenbolbidex = s(3,3{1,1,2}1{1,1,2}2)

And adding “-plex” after “-biplex”:

  • dimensolbidexplex = s(3,3{2,1,2}2)
  • dimentrienbidexplex = s(3,3,3{2,1,2}2)
  • dimentrilbidexplex = s(3,3{2,1,2}3)
  • dimenbolbidexplex = s(3,3{2,1,2}1{2,1,2}2)

We can also add “-plex” prior to the “-biplex”, but this step is too big. The “-bidex” means two “-dex”es, so we can insert “-plex” between them, such as

  • dimensoldexplexdex = s(3,3{1,2,2}2) = s(3,3{3,1,2}2)
  • dimentriendexplexdex = s(3,3,3{1,2,2}2)
  • dimentrildexplexdex = s(3,3{1,2,2}3)
  • dimenboldexplexdex = s(3,3{1,2,2}1{1,2,2}2)
  • dimensoldexplexdexplex = s(3,3{2,2,2}2)
  • dimentriendexplexdexplex = s(3,3,3{2,2,2}2)
  • dimentrildexplexdexplex = s(3,3{2,2,2}3)
  • dimenboldexplexdexplex = s(3,3{2,2,2}1{2,2,2}2)
  • dimentriendexbiplexdex = s(3,3,3{1,3,2}2)
  • dimentrildexbiplexdex = s(3,3{1,3,2}3)
  • dimenboldexbiplexdex = s(3,3{1,3,2}1{1,3,2}2)

Then we reach “-plex” prior to “-bidex”.

  • dimensolthrex = s(3,3{1{2}2}2) = s(3,3{1,1,1,2}2) = s(3,3{1,1,3}2) = s(3,3{1,3,2}2) = s(3,3{3,2,2}2), a.k.a. dimensoltridex, dimensolplexbidex, dimensoldexbiplexdex and dimensoldexplexdexbiplex.
  • dimentrienplexbidex = s(3,3,3{1,1,3}2)
  • dimentrilplexbidex = s(3,3{1,1,3}3)
  • dimenbolplexbidex = s(3,3{1,1,3}1{1,1,3}2)
  • dimensolplexbidexplex = s(3,3{2,1,3}2)
  • dimentrienplexbidexplex = s(3,3,3{2,1,3}2)
  • dimentrilplexbidexplex = s(3,3{2,1,3}3)
  • dimenbolplexbidexplex = s(3,3{2,1,3}1{2,1,3}2)
  • dimensolplexdexplexdex = s(3,3{1,2,3}2) = s(3,3{3,1,3}2)
  • dimentrienplexdexplexdex = s(3,3,3{1,2,3}2)
  • dimentrilplexdexplexdex = s(3,3{1,2,3}3)
  • dimenbolplexdexplexdex = s(3,3{1,2,3}1{1,2,3}2)
  • dimentrienplexdexbiplexdex = s(3,3,3{1,3,3}2)
  • dimentrilplexdexbiplexdex = s(3,3{1,3,3}3)
  • dimenbolplexdexbiplexdex = s(3,3{1,3,3}1{1,3,3}2)

And with “-biplex”:

  • dimensolbiplexbidex = s(3,3{1,1,4}2) = s(3,3{1,3,3}2) = s(3,3{3,2,3}2), a.k.a. dimensolplexdexbiplexdex.
  • dimentrienbiplexbidex = s(3,3,3{1,1,4}2)
  • dimentrilbiplexbidex = s(3,3{1,1,4}3)
  • dimenbolbiplexbidex = s(3,3{1,1,4}1{1,1,4}2)

And so on. The “n-plex” prior to the “-bidex” means adding n to the 3rd entry of the separator A.

Multi-dex group

The suffix “-tridex” means 3 “-dex”es, so we can insert “-plex”es before them, after one, after two, or after them, such as

  • dimentrientridex = s(3,3,3{1,1,1,2}2)
  • dimentriltridex = s(3,3{1,1,1,2}3)
  • dimenboltridex = s(3,3{1,1,1,2}1{1,1,1,2}2)
  • dimensoltridexplex = s(3,3{2,1,1,2}2)
  • dimensolbidexplexdex = s(3,3{1,2,1,2}2) = s(3,3{3,1,1,2}2)
  • dimensolbidexplexdexplex = s(3,3{2,2,1,2}2)
  • dimensoldexplexbidex = s(3,3{1,1,2,2}2) = s(3,3{1,3,1,2}2) = s(3,3{3,2,1,2}2)
  • dimensoldexplexbidexplex = s(3,3{2,1,2,2}2)
  • dimensoldexplexdexplexdex = s(3,3{1,2,2,2}2) = s(3,3{3,1,2,2}2)
  • dimensoldexplexdexplexdexplex = s(3,3{2,2,2,2}2)
  • dimensolquadridex = s(3,3{1,1,1,1,2}2) = s(3,3{1,1,1,3}2) = s(3,3{1,1,3,2}2) = s(3,3{1,3,2,2}2) = s(3,3{3,2,2,2}2), a.k.a. dimensolplextridex.
  • dimensolbiplextridex = s(3,3{1,1,1,4}2) = s(3,3{1,1,3,3}2) = s(3,3{1,3,2,3}2) = s(3,3{3,2,2,3}2)

The suffix “-quadridex” means 4 “-dex”es, so we can also insert “-plex”es.

  • dimentrienquadridex = s(3,3,3{1,1,1,1,2}2)
  • dimentrilquadridex = s(3,3{1,1,1,1,2}3)
  • dimenbolquadridex = s(3,3{1,1,1,1,2}1{1,1,1,1,2}2)
  • dimensolquadridexplex = s(3,3{2,1,1,1,2}2)
  • dimensoltridexplexdex = s(3,3{1,2,1,1,2}2) = s(3,3{3,1,1,1,2}2)
  • dimensoltridexplexdexplex = s(3,3{2,2,1,1,2}2)
  • dimensolbidexplexbidex = s(3,3{1,1,2,1,2}2) = s(3,3{1,3,1,1,2}2) = s(3,3{3,2,1,1,2}2)
  • dimensolbidexplexbidexplex = s(3,3{2,1,2,1,2}2)
  • dimensolbidexplexdexplexdex = s(3,3{1,2,2,1,2}2) = s(3,3{3,1,2,1,2}2)
  • dimensolbidexplexdexplexdexplex = s(3,3{2,2,2,1,2}2)
  • dimensoldexplextridex = s(3,3{1,1,1,2,2}2) = s(3,3{1,1,3,1,2}2) = s(3,3{1,3,2,1,2}2) = s(3,3{3,2,2,1,2}2)
  • dimensoldexplextridexplex = s(3,3{2,1,1,2,2}2)
  • dimensoldexplexbidexplexdex = s(3,3{1,2,1,2,2}2) = s(3,3{3,1,1,2,2}2)
  • dimensoldexplexbidexplexdexplex = s(3,3{2,2,1,2,2}2)
  • dimensoldexplexdexplexbidex = s(3,3{1,1,2,2,2}2) = s(3,3{1,3,1,2,2}2) = s(3,3{3,2,1,2,2}2)
  • dimensoldexplexdexplexbidexplex = s(3,3{2,1,2,2,2}2)
  • dimensoldexplexdexplexdexplexdex = s(3,3{1,2,2,2,2}2) = s(3,3{3,1,2,2,2}2)
  • dimensoldexplexdexplexdexplexdexplex = s(3,3{2,2,2,2,2}2)
  • dimensolquintidex = s(3,3{1,1,1,1,1,2}2) = s(3,3{1,1,1,1,3}2) = s(3,3{1,1,1,3,2}2) = s(3,3{1,1,3,2,2}2) = s(3,3{1,3,2,2,2}2) = s(3,3{3,2,2,2,2}2), a.k.a. dimensolplexquadridex.
  • dimensolbiplexquadridex = s(3,3{1,1,1,1,4}2) = s(3,3{1,1,1,3,3}2) = s(3,3{1,1,3,2,3}2) = s(3,3{1,3,2,2,3}2) = s(3,3{3,2,2,2,3}2)

And the suffix “-quintidex” means 5 “-dex”es.

  • dimentrienquintidex = s(3,3,3{1,1,1,1,1,2}2)
  • dimentrilquintidex = s(3,3{1,1,1,1,1,2}3)
  • dimenbolquintidex = s(3,3{1,1,1,1,1,2}1{1,1,1,1,1,2}2)
  • dimensolquintidexplex = s(3,3{2,1,1,1,1,2}2)
  • dimensolquadridexplexdex = s(3,3{1,2,1,1,1,2}2) = s(3,3{3,1,1,1,1,2}2)
  • dimensoltridexplexbidex = s(3,3{1,1,2,1,1,2}2) = s(3,3{1,3,1,1,1,2}2) = s(3,3{3,2,1,1,1,2}2)
  • dimensolbidexplextridex = s(3,3{1,1,1,2,1,2}2) = s(3,3{1,1,3,1,1,2}2) = s(3,3{1,3,2,1,1,2}2) = s(3,3{3,2,2,1,1,2}2)
  • dimensoldexplexquadridex = s(3,3{1,1,1,1,2,2}2) = s(3,3{1,1,1,3,1,2}2) = s(3,3{1,1,3,2,1,2}2) = s(3,3{1,3,2,2,1,2}2) = s(3,3{3,2,2,2,1,2}2)
  • dimensolplexquintidex = s(3,3{1,1,1,1,1,3}2) = s(3,3{1,1,1,1,3,2}2) = s(3,3{1,1,1,3,2,2}2) = s(3,3{1,1,3,2,2,2}2) = s(3,3{1,3,2,2,2,2}2) = s(3,3{3,2,2,2,2,2}2). Also s(3,3{1,1,1,1,1,1,2}2) = s(3,3{1,1,1,1,1,3}2) = dimensolplexquintidex.

And so on. Now have a look at effects of suffix “-dex” and “-plex”. First, both modify the “separator A” in the definition of namebases. The “n-dex” increase the (n+1)-th entry of A from 1 to 2, then the “n-plex” prior to m “-dex”es increase the (m+1)-th entry of A by n. However, since we start on A = {1,2}, names below dimensolbidex have an offset on the 2nd entry of A.

Naming space

Now I introduce an idea about naming numbers – the naming space. The size of naming space of a naming system shows how many regular names the system can produce.

Here’s how “-plex” and “-dex” works in my naming system. We can have arbitrary many “-plex”es, and shorten as “n-plex”, so this forms a naming space of {\omega}. Notice that I don’t write {\aleph_0} because the levels of “-plex”, “-biplex”, “-triplex”, etc. are ordered. We want “-biplex” to be a higher-leveled modifier than “-plex”, and “-triplex” is even higher-leveled, and so on.

Next, a “-dex” is a stronger modifier than all the “n-plex”es, and “-dex” followed by “n-plex” form another naming space of {\omega}, and of {\omega2} in total. Then “-plexdex” followed by “n-plex” form another {\omega}, and so on. So “n-plexdex-m-plex” form a naming space of {\omega^2}. We still want them to be ordered, e.g. “-plexdex” is higher-leveled than all the “-dex-n-plex”es.

Next, 2 “-dex”es separate “-plex”es into 3 groups, so they form a naming space of {\omega^3}; 3 “-dex”es separate “-plex”es into 4 groups, so they form a naming space of {\omega^4}, and so on. Multi-dex and “-plex”es then form a naming space of {\omega^\omega}.

What about the notation system? With “-plex”es and “-dex”es I named numbers up to separator {1{2}2}, which has recursion level {\omega^\omega} – that’s equal to the size of the naming space, so we can have a name for numbers at every separator recursion level. However, function f(n) = s(n,n{1{2}2}2) has growth rate {\omega^{\omega^{\omega^\omega}}>\omega^\omega}, so the naming space is not enough if I must name for every growth rate (in FGH scale). I indeed named numbers up to separator {1{2}2}, but you can see what’s wrong with the naming system. The problem is that there’re many naming holes. e.g. between two adjacent terms in the naming system, dimentrienquintidex (where s(n,n,3{1,1,1,1,1,2}2) has growth rate {\omega^{\omega^{\omega^5}}+2}) and dimentrilquintidex (where s(n,n{1,1,1,1,1,2}3) has growth rate {\omega^{\omega^{\omega^5}}2}), there’re many “unnamed growth rate” (i.e. without any named numbers), so they form a naming hole.

So how to deal with naming holes? One way is to leave it empty, but we can’t feel the hugeness of the step from before the hole to after the hole – it’s not good. A second way is to change the order of the modifiers we use, to form a larger naming space, then it fit our need. But, the new order may be more chaotic, and the result names may be longer and unclear. That’s a “notational way”, with more and more rules, not suit for naming. Another way is to add new modifiers to the naming system, which can fill the naming holes. Then the types of modifiers will be greater as we go on. That’s a “naming way”, with more and more new modifiers, and we get more and more colorful names – that’s good. So I choose the last one for the next steps.

-threx group

Now I introduce a new modifier – the suffix “-threx”. It means the 3rd single modifier after “-plex” and “-dex”. Here’s the basic framework of the naming system. First, we have “n-threx”, meaning n “-threx”es, and this form a naming space of {\omega}. Second, we insert “-dex”es between “-threx”es, and this step multiply the naming space by {\omega^\omega}. Finally, we insert “-plex”es between suffixes already existing, and this step multiply the naming space by {\omega^\omega} again. So the size of the naming space is {\omega^{\omega2}}. However, I still want the system can go up to separator recursion level {\omega^{\omega^\omega}}! So I must need more modifiers…

Let’s begin with a single “-threx”.

  • dimentrienthrex = s(3,3,3{1{2}2}2)
  • dimentrilthrex = s(3,3{1{2}2}3)
  • dimenbolthrex = s(3,3{1{2}2}1{1{2}2}2)

Then add “-plex”es after the “-threx”.

  • dimensolthrexplex = s(3,3{2{2}2}2)
  • dimentrienthrexplex = s(3,3,3{2{2}2}2)
  • dimentrilthrexplex = s(3,3{2{2}2}3)
  • dimenbolthrexplex = s(3,3{2{2}2}1{2{2}2}2)
  • dimensolthrexbiplex = s(3,3{3{2}2}2)
  • dimentrienthrexbiplex = s(3,3,3{3{2}2}2)
  • dimentrilthrexbiplex = s(3,3{3{2}2}3)
  • dimenbolthrexbiplex = s(3,3{3{2}2}1{3{2}2}2)

Then we can add “-plex”es prior to the “-threx”, such as

  • dimentrienplexthrex = s(3,3,3{1{2}3}2)
  • dimentrilplexthrex = s(3,3{1{2}3}3)
  • dimenbolplexthrex = s(3,3{1{2}3}1{1{2}3}2)

But wait! There’s a naming hole between {1,2{2}2} and {1{2}3}. To fill the naming hole, I use a new modifier.

  • dimensolthrexdexiplex = s(3,3{1,2{2}2}2) = s(3,3{3,1{2}2}2), which is slightly larger than dimensolthrexbiplex.
  • dimentrienthrexdexiplex = s(3,3,3{1,2{2}2}2)
  • dimentrilthrexdexiplex = s(3,3{1,2{2}2}3)
  • dimenbolthrexdexiplex = s(3,3{1,2{2}2}1{1,2{2}2}2)
  • dimensolthrexdexibiplex = s(3,3{2,2{2}2}2)
  • dimensolthrexbidexiplex = s(3,3{1,1,2{2}2}2) = s(3,3{1,3,1{2}2}2)
  • dimensolthrexbidexibiplex = s(3,3{2,1,2{2}2}2)
  • dimensolthrexdexiplexdexiplex = s(3,3{1,2,2{2}2}2) = s(3,3{3,1,2{2}2}2), a.k.a. dimensolthrexbidexitriplex.
  • dimensolthrexdexiplexdexibiplex = s(3,3{2,2,2{2}2}2)
  • dimensolthrexdex = s(3,3{1{2}1,2}2) = s(3,3{1{2}3}2) = s(3,3{1,1,1,2{2}2}2), a.k.a. dimensolplexthrex and dimensolthrextridexiplex.

Notice that I use “-dexi” instead of “-dex”, and I always leave one more “-plex” at the end of the name. That’s my way to fill the naming hole. Now continue upward…

  • dimensolplexthrexplex = s(3,3{2{2}3}2)
  • dimensolbiplexthrex = s(3,3{1{2}4}2) = s(3,3{1,1,1,2{2}3}2) = s(3,3{1,1,3,1{2}3}2) – and we have the same way to fill the naming hole between the first two numbers.
  • dimensolthrexdexplex = s(3,3{2{2}1,2}2)
  • dimensolthrexdexdexiplex = s(3,3{1,2{2}1,2}2)
  • dimensolthrexplexdex = s(3,3{1{2}2,2}2)
  • dimensolthrexplexdexplex = s(3,3{2{2}2,2}2)
  • dimensolplexthrexdex = s(3,3{1{2}1,3}2). Also dimensolthrexbidex = s(3,3{1{2}1,1,2}2) = s(3,3{1{2}1,3}2) = dimensolplexthrexdex.
  • dimensolplexthrexdexplex = s(3,3{2{2}1,3}2)
  • dimensolplexthrexplexdex = s(3,3{1{2}2,3}2)
  • dimensolplexthrexplexdexplex = s(3,3{2{2}2,3}2)

A “-threx” and a “-dex” can form 3 places where “n-plex” can insert. The last place corresponds to increasing the 1st entry of {1{2}1,2}, the 2nd place for the 2nd entry of {1{2}1,2}, and the 1st place for the last entry. For more “-threx”es and “-dex”es, this still holds.

  • dimensolthrexbidexplex = s(3,3{2{2}1,1,2}2)
  • dimensolthrexdexplexdex = s(3,3{1{2}2,1,2}2)
  • dimensolthrexplexbidex = s(3,3{1{2}1,2,2}2)
  • dimensoldexthrex = s(3,3{1{2}1{2}2}2) = s(3,3{1{2}1,1,1,2}2) = s(3,3{1{2}1,1,3}2), a.k.a. dimensolthrextridex and dimensolplexthrexbidex.
  • dimensoldexthrexplex = s(3,3{2{2}1{2}2}2)
  • dimensoldexplexthrex = s(3,3{1{2}2{2}2}2)
  • dimensoldexthrexdex = s(3,3{1{2}1{2}1,2}2) = s(3,3{1{2}1{2}3}2), a.k.a. dimensolplexdexthrex .
  • dimensoldexthrexdexplex = s(3,3{2{2}1{2}1,2}2)
  • dimensoldexthrexplexdex = s(3,3{1{2}2{2}1,2}2)
  • dimensoldexplexthrexdex = s(3,3{1{2}1{2}2,2}2)
  • dimensoldexthrexbidex = s(3,3{1{2}1{2}1,1,2}2) = s(3,3{1{2}1{2}1,3}2), a.k.a. dimensolplexdexthrexdex.
  • dimensoldexthrexbidexplex = s(3,3{2{2}1{2}1,1,2}2)
  • dimensoldexthrexdexplexdex = s(3,3{1{2}2{2}1,1,2}2)
  • dimensoldexthrexplexbidex = s(3,3{1{2}1{2}2,1,2}2)
  • dimensoldexplexthrexbidex = s(3,3{1{2}1{2}1,2,2}2)
  • extendol = s(3,3{1`2}2) = s(3,3{1{1,2}2}2) = s(3,3{1{3}2}2) = s(3,3{1{2}1{2}1{2}2}2) = s(3,3{1{2}1{2}1,1,1,2}2) = s(3,3{1{2}1{2}1,1,3}2), a.k.a. dimensoltetrex, dimensolbithrex, dimensolbidexthrex, dimensoldexthrextridex and dimensolplexdexthrexbidex. For more information about this number, see a later page “numbers from expanding array notation”.

Generally, a “-dex” add an entry at the end of the layer-1 separator (i.e. the “separator A”). If you want some expressions with extra entry not at the end of the separator A, then you go into a naming hole, and you need “-dexi”es. e.g. dimensoldexthrexdexiplexdex = s(3,3{1{2}1,2{2}1,2}2) = s(3,3{1{2}3,1{2}1,2}2).

  • dimensolbithrexplex = s(3,3{2{3}2}2)
  • dimensolthrexplexthrex = s(3,3{1{2}2{3}2}2). Hey! There’s an extra entry! This entry corresponds to {1{2}2}, or the strength of a “-threx”.
  • dimensolthrexdexiplexthrex = s(3,3{1{2}1,2{3}2}2) – for other extra entries we need “-dexi”es.
  • dimensolthrexthrexiplexthrex = s(3,3{1{2}1{2}2{3}2}2) – now “-threxi” appears.
  • dimensolthrexthrexidexiiplexthrex = s(3,3{1{2}1,2{2}2{3}2}2) – now “-dexii” appears. That’s a way to fill an “2nd-order” naming hole, where even with “-dexi” and “-threxi” we still can’t cover. If there’re still naming holes, we’ll use more Roman numerals such as “-dexiii”, “-dexiv”, “-dexv”, “-dexvi”, etc.
  • dimensolbithrexdex = s(3,3{1{3}1,2}2) = s(3,3{1{3}3}2), a.k.a. dimensolplexbithrex.
  • dimensolbithrexdexplex = s(3,3{2{3}1,2}2)
  • dimensolbithrexplexdex = s(3,3{1{2}2{3}1,2}2)
  • dimensolthrexplexthrexdex = s(3,3{1{3}2,2}2)
  • dimensolbithrexbidex = s(3,3{1{3}1,1,2}2) = s(3,3{1{3}1,3}2), a.k.a. dimensolplexbithrexdex.
  • dimensolthrexdexthrex = s(3,3{1{3}1{2}2}2) = s(3,3{1{3}1,1,1,2}2) = s(3,3{1{3}1,1,3}2), a.k.a. dimensolbithrextridex and dimensolplexbithrexbidex.
  • dimensolthrexdexthrexdex = s(3,3{1{3}1{2}1,2}2)
  • dimensolthrexbidexthrex = s(3,3{1{3}1{2}1{2}2}2)
  • dimensoldexbithrex = s(3,3{1{3}1{3}2}2) = s(3,3{1{3}1{2}1{2}1{2}2}2), a.k.a. dimensolthrextridexthrex.
  • dimensoltrithrex = s(3,3{1{4}2}2) = s(3,3{1{3}1{3}1{3}2}2), a.k.a. dimensolbidexbithrex.
  • dimensoltrithrexplex = s(3,3{2{4}2}2)
  • dimensolbithrexplexthrex = s(3,3{1{2}2{4}2}2)
  • dimensolthrexplexbithrex = s(3,3{1{3}2{4}2}2)
  • dimensoltrithrexdex = s(3,3{1{4}1,2}2) = s(3,3{1{4}3}2), a.k.a. dimensolplextrithrex.
  • dimensolbithrexdexthrex = s(3,3{1{4}1{2}2}2)
  • dimensolthrexdexbithrex = s(3,3{1{4}1{3}2}2)
  • dimensoldextrithrex = s(3,3{1{4}1{4}2}2)
  • dimensolquadrithrex = s(3,3{1{5}2}2) = s(3,3{1{4}1{4}1{4}2}2), a.k.a. dimensolbidextrithrex.
  • dimensolquintithrex = s(3,3{1{6}2}2)

Now I show you how the main naming system (excluding the filling naming hole part) works for the “-threx” level.

  1. At the beginning, 4 namebases are mapped into expressions. Separator A is initialized as {1,2}.
  2. Append “-threx”es to the name. The entry inside the separator inside A increases 1 every appending.
  3. Insert “-dex”es to the name. For every insertion, look at the position from the end of the name. If it’s the k-th position, the A will add an entry at the end, where the separator corresponds to the result of “appending k ‘-threx’es at step 2”.
  4. Insert “-plex”es to the name. For every insertion, look at the position from the end of the name. If it’s the k-th position, the k-th entry of A will increases 1.
  5. Replace multiple same suffixes with “bi-“, “tri-“, “quadri-” etc. ones.

Using ordinals, the naming process can be more clear. A “-threx” means {\omega^{\omega^\alpha}\mapsto\omega^{\omega^{\alpha+\beta}}}, a “-dex” means {\omega^\alpha\mapsto\omega^{\alpha+\beta}}, and a “-plex” means {\alpha\mapsto\alpha+\beta} in separator recursion level, where the β is defined by the result of adding k suffixes in previous steps, where k is the position of the suffix we focus from the end of the name. What’s more important, this way works even beyond the “-threx” group!

-tetrex and beyond

Using suffix “-tetrex”, we have a naming space of {\omega^{\omega3}}.

  • dimentrientetrex = s(3,3,3{1{1,2}2}2)
  • dimentriltetrex = s(3,3{1{1,2}2}3)
  • dimenboltetrex = s(3,3{1{1,2}2}1{1{1,2}2}2)
  • dimensoltetrexplex = s(3,3{2{1,2}2}2)
  • dimensoltetrexdexiplex = s(3,3{1,2{1,2}2}2) = s(3,3{3,1{1,2}2}2)
  • dimensoltetrexplexdexiplex = s(3,3{1,3{1,2}2}2)
  • dimensoltetrexbidexiplex = s(3,3{1,1,2{1,2}2}2) = s(3,3{1,3,1{1,2}2}2)
  • dimensoltetrexthrexiplex = s(3,3{1{2}2{1,2}2}2) = s(3,3{1,1,1,2{2}1{1,2}2}2)
  • dimensoltetrexthrexidexiiplex = s(3,3{1,2{2}2{1,2}2}2) = s(3,3{3,1{2}2{1,2}2}2)
  • dimensoltetrexthrexidexiplex = s(3,3{1{2}1,2{1,2}2}2) = s(3,3{1{2}3,1{1,2}2}2)
  • dimensoltetrexdexithrexiplex = s(3,3{1{2}1{2}2{1,2}2}2) = s(3,3{1{2}1,1,1,2{2}1{1,2}2}2)
  • dimensoltetrexdex = s(3,3{1{1,2}1,2}2) = s(3,3{1{1,2}3}2) = s(3,3{1{3}2{1,2}2}2), a.k.a. dimensolplextetrex and dimensoltetrexbithrexiplex.
  • dimensoltetrexdexplex = s(3,3{2{1,2}1,2}2)
  • dimensoltetrexplexdex = s(3,3{1{1,2}2,2}2)
  • dimensoltetrexbidex = s(3,3{1{1,2}1,1,2}2) = s(3,3{1{1,2}1,3}2), a.k.a. dimensolplextetrexdex.
  • dimensoltetrexthrexidex = s(3,3{1{1,2}1{2}2}2) – Now a “-threxi” appears prior to a “-dex”.
  • dimensoldextetrex = s(3,3{1{1,2}1{1,2}2}2) = s(3,3{1{1,2}1{3}2}2), a.k.a. dimensoltetrexbithrexidex.
  • dimensoldextetrexplex = s(3,3{2{1,2}1{1,2}2}2)
  • dimensoldexplextetrex = s(3,3{1{1,2}2{1,2}2}2)
  • dimensoldextetrexdex = s(3,3{1{1,2}1{1,2}1,2}2) = s(3,3{1{1,2}1{1,2}3}2), a.k.a. dimensolplexdextetrex.
  • dimensoltetrexthrex = s(3,3{1{2,2}2}2) = s(3,3{1{1,2}1{1,2}1{1,2}2}2), a.k.a. dimensolbidextetrex.
  • dimensoltetrexthrexplex = s(3,3{2{2,2}2}2)
  • dimensoltetrexplexthrex = s(3,3{1{1,2}2{2,2}2}2)
  • dimensoltetrexthrexdex = s(3,3{1{2,2}1,2}2) = s(3,3{1{2,2}3}2), a.k.a. dimensolplextetrexthrex.
  • dimensoltetrexthrexdexplex = s(3,3{2{2,2}1,2}2)
  • dimensoltetrexthrexplexdex = s(3,3{1{1,2}2{2,2}1,2}2)
  • dimensoltetrexplexthrexdex = s(3,3{1{2,2}2,2}2)
  • dimensoltetrexthrexbidex = s(3,3{1{2,2}1,1,2}2) = s(3,3{1{2,2}1,3}2), a.k.a. dimensolplextetrexthrexdex.
  • dimensoltetrexdexthrex = s(3,3{1{2,2}1{1,2}2}2)
  • dimensoltetrexdexthrexplex = s(3,3{2{2,2}1{1,2}2}2)
  • dimensoltetrexdexplexthrex = s(3,3{1{1,2}2{2,2}1{1,2}2}2)
  • dimensoltetrexplexdexthrex = s(3,3{1{2,2}2{1,2}2}2)
  • dimensoltetrexdexthrexdex = s(3,3{1{2,2}1{1,2}1,2}2) = s(3,3{1{2,2}1{1,2}3}2), a.k.a. dimensolplextetrexdexthrex.
  • dimensoltetrexdexthrexbidex = s(3,3{1{2,2}1{1,2}1,1,2}2)
  • dimensoltetrexbidexthrex = s(3,3{1{2,2}1{1,2}1{1,2}2}2)
  • dimensoldextetrexthrex = s(3,3{1{2,2}1{2,2}2}2)
  • dimensoldextetrexthrexdex = s(3,3{1{2,2}1{2,2}1,2}2)
  • dimensoldextetrexthrexbidex = s(3,3{1{2,2}1{2,2}1,1,2}2)
  • dimensoldextetrexdexthrex = s(3,3{1{2,2}1{2,2}1{1,2}2}2)
  • dimensoldextetrexdexthrexdex = s(3,3{1{2,2}1{2,2}1{1,2}1,2}2)
  • dimensoldextetrexdexthrexbidex = s(3,3{1{2,2}1{2,2}1{1,2}1,1,2}2)
  • dimensoldextetrexbidexthrex = s(3,3{1{2,2}1{2,2}1{1,2}1{1,2}2}2)
  • dimensolbitetrex = s(3,3{1{1,1,2}2}2) = s(3,3{1{1,3}2}2) = s(3,3{1{3,2}2}2) = s(3,3{1{2,2}1{2,2}1{2,2}2}2), a.k.a. dimensolthrextetrex, dimensoltetrexbithrex and dimensolbidextetrexthrex.
  • dimensolbitetrexplex = s(3,3{2{1,1,2}2}2)
  • dimensoltetrexplextetrex = s(3,3{1{1,2}2{1,1,2}2}2)
  • dimensolbitetrexdex = s(3,3{1{1,1,2}1,2}2) = s(3,3{1{1,1,2}3}2), a.k.a. dimensolplexbitetrex.
  • dimensolbitetrexdexplex = s(3,3{2{1,1,2}1,2}2)
  • dimensolbitetrexplexdex = s(3,3{1{1,2}2{1,1,2}1,2}2)
  • dimensoltetrexplextetrexdex = s(3,3{1{1,1,2}2,2}2)
  • dimensolbitetrexbidex = s(3,3{1{1,1,2}1,1,2}2) = s(3,3{1{1,1,2}1,3}2), a.k.a. dimensolplexbitetrexdex.
  • dimensoltetrexdextetrex = s(3,3{1{1,1,2}1{1,2}2}2)
  • dimensoltetrexdextetrexplex = s(3,3{2{1,1,2}1{1,2}2}2)
  • dimensoltetrexdexplextetrex = s(3,3{1{1,2}2{1,1,2}1{1,2}2}2)
  • dimensoltetrexplexdextetrex = s(3,3{1{1,1,2}2{1,2}2}2)
  • dimensoltetrexdextetrexdex = s(3,3{1{1,1,2}1{1,2}1,2}2) = s(3,3{1{1,1,2}1{1,2}3}2), a.k.a. dimensolplextetrexdextetrex.
  • dimensoltetrexdextetrexbidex = s(3,3{1{1,1,2}1{1,2}1,1,2}2)
  • dimensoltetrexbidextetrex = s(3,3{1{1,1,2}1{1,2}1{1,2}2}2)
  • dimensoltetrexbidextetrexdex = s(3,3{1{1,1,2}1{1,2}1{1,2}1,2}2)
  • dimensoltetrexbidextetrexbidex = s(3,3{1{1,1,2}1{1,2}1{1,2}1,1,2}2)
  • dimensoldexbitetrex = s(3,3{1{1,1,2}1{1,1,2}2}2)
  • dimensoldexbitetrexplex = s(3,3{2{1,1,2}1{1,1,2}2}2)
  • dimensoldextetrexplextetrex = s(3,3{1{1,2}2{1,1,2}1{1,1,2}2}2)
  • dimensoldexplexbitetrex = s(3,3{1{1,1,2}2{1,1,2}2}2)
  • dimensoldexbitetrexdex = s(3,3{1{1,1,2}1{1,1,2}1,2}2) = s(3,3{1{1,1,2}1{1,1,2}3}2), a.k.a. dimensolplexdexbitetrex.
  • dimensoldexbitetrexbidex = s(3,3{1{1,1,2}1{1,1,2}1,1,2}2)
  • dimensoldextetrexdextetrex = s(3,3{1{1,1,2}1{1,1,2}1{1,2}2}2)
  • dimensoldextetrexbidextetrex = s(3,3{1{1,1,2}1{1,1,2}1{1,2}1{1,2}2}2)
  • dimensolbitetrexthrex = s(3,3{1{2,1,2}2}2) = s(3,3{1{1,1,2}1{1,1,2}1{1,1,2}2}2), a.k.a. dimensolbidexbitetrex.
  • dimensolbitetrexthrexplex = s(3,3{2{2,1,2}2}2)
  • dimensolbitetrexplexthrex = s(3,3{1{1,2}2{2,1,2}2}2)
  • dimensoltetrexplextetrexthrex = s(3,3{1{1,1,2}2{2,1,2}2}2)
  • dimensolbitetrexthrexdex = s(3,3{1{2,1,2}1,2}2) = s(3,3{1{2,1,2}3}2), a.k.a. dimensolplexbitetrexthrex.
  • dimensolbitetrexthrexbidex = s(3,3{1{2,1,2}1,1,2}2)
  • dimensolbitetrexdexthrex = s(3,3{1{2,1,2}1{1,2}2}2)
  • dimensolbitetrexdexthrexdex = s(3,3{1{2,1,2}1{1,2}1,2}2)
  • dimensolbitetrexdexthrexbidex = s(3,3{1{2,1,2}1{1,2}1,1,2}2)
  • dimensolbitetrexbidexthrex = s(3,3{1{2,1,2}1{1,2}1{1,2}2}2)
  • dimensoltetrexdextetrexthrex = s(3,3{1{2,1,2}1{1,1,2}2}2)
  • dimensoltetrexdextetrexdexthrex = s(3,3{1{2,1,2}1{1,1,2}1{1,2}2}2)
  • dimensoltetrexdextetrexbidexthrex = s(3,3{1{2,1,2}1{1,1,2}1{1,2}1{1,2}2}2)
  • dimensoltetrexbidextetrexthrex = s(3,3{1{2,1,2}1{1,1,2}1{1,1,2}2}2)
  • dimensoldexbitetrexthrex = s(3,3{1{2,1,2}1{2,1,2}2}2)
  • dimensoldexbitetrexthrexdex = s(3,3{1{2,1,2}1{2,1,2}1,2}2)
  • dimensoldexbitetrexdexthrex = s(3,3{1{2,1,2}1{2,1,2}1{1,2}2}2)
  • dimensoldextetrexdextetrexthrex = s(3,3{1{2,1,2}1{2,1,2}1{1,1,2}2}2)
  • dimensoltetrexthrextetrex = s(3,3{1{1,2,2}2}2) = s(3,3{1{3,1,2}2}2) = s(3,3{1{2,1,2}1{2,1,2}1{2,1,2}2}2), a.k.a. dimensolbitetrexbithrex and dimensolbidexbitetrexthrex.
  • dimensoltetrexthrextetrexdex = s(3,3{1{1,2,2}1,2}2)
  • dimensoltetrexthrexdextetrex = s(3,3{1{1,2,2}1{1,2}2}2)
  • dimensoltetrexdexthrextetrex = s(3,3{1{1,2,2}1{1,1,2}2}2)
  • dimensoldextetrexthrextetrex = s(3,3{1{1,2,2}1{1,2,2}2}2)
  • dimensoltetrexthrextetrexthrex = s(3,3{1{2,2,2}2}2)
  • dimensolpentex = s(3,3{1{1{2}2}2}2) = s(3,3{1{1,1,1,2}2}2) = s(3,3{1{1,1,3}2}2) = s(3,3{1{1,3,2}2}2) = s(3,3{1{3,2,2}2}2), a.k.a. dimensoltritetrex, dimensolthrexbitetrex, dimensoltetrexbithrextetrex and dimensoltetrexthrextetrexbithrex.
  • dimensoltritetrexplex = s(3,3{2{1,1,1,2}2}2)
  • dimensolbitetrexplextetrex = s(3,3{1{1,2}2{1,1,1,2}2}2)
  • dimensoltetrexplexbitetrex = s(3,3{1{1,1,2}2{1,1,1,2}2}2)
  • dimensoltritetrexdex = s(3,3{1{1,1,1,2}1,2}2) = s(3,3{1{1,1,1,2}3}2), a.k.a. dimensolplextritetrex.
  • dimensolbitetrexdextetrex = s(3,3{1{1,1,1,2}1{1,2}2}2)
  • dimensoltetrexdexbitetrex = s(3,3{1{1,1,1,2}1{1,1,2}2}2)
  • dimensoldextritetrex = s(3,3{1{1,1,1,2}1{1,1,1,2}2}2)
  • dimensoltritetrexthrex = s(3,3{1{2,1,1,2}2}2)
  • dimensolbitetrexthrextetrex = s(3,3{1{1,2,1,2}2}2)
  • dimensoltetrexthrexbitetrex = s(3,3{1{1,1,2,2}2}2)
  • dimensolquadritetrex = s(3,3{1{1,1,1,1,2}2}2) = s(3,3{1{1,1,1,3}2}2), a.k.a. dimensolthrextritetrex.
  • dimensolquadritetrexplex = s(3,3{2{1,1,1,1,2}2}2)
  • dimensolquadritetrexdex = s(3,3{1{1,1,1,1,2}1,2}2) = s(3,3{1{1,1,1,1,2}3}2), a.k.a. dimensolplexquadritetrex.
  • dimensoldexquadritetrex = s(3,3{1{1,1,1,1,2}1{1,1,1,1,2}2}2)
  • dimensolquadritetrexthrex = s(3,3{1{2,1,1,1,2}2}2)
  • dimensolquintitetrex = s(3,3{1{1,1,1,1,1,2}2}2) = s(3,3{1{1,1,1,1,3}2}2), a.k.a. dimensolthrexquadritetrex.

Using suffix “-pentex”, we have a naming space of {\omega^{\omega4}}.

  • dimentrienpentex = s(3,3,3{1{1{2}2}2}2)
  • dimentrilpentex = s(3,3{1{1{2}2}2}3)
  • dimenbolpentex = s(3,3{1{1{2}2}2}1{1{1{2}2}2}2)
  • dimensolpentexplex = s(3,3{2{1{2}2}2}2)
  • dimensolpentexdex = s(3,3{1{1{2}2}1,2}2) = s(3,3{1{1{2}2}3}2), a.k.a. dimensolplexpentex.
  • dimensoldexpentex = s(3,3{1{1{2}2}1{1{2}2}2}2)
  • dimensolpentexthrex = s(3,3{1{2{2}2}2}2)
  • dimensolpentextetrex = s(3,3{1{1{2}1,2}2}2) = s(3,3{1{1{2}3}2}2), a.k.a. dimensolthrexpentex.
  • dimensoltetrexpentex = s(3,3{1{1{2}1{2}2}2}2)
  • dimensolhex = s(3,3{1{1{1,2}2}2}2) = s(3,3{1{1{3}2}2}2), a.k.a. dimensolbipentex.
  • dimensolbipentexplex = s(3,3{2{1{3}2}2}2)
  • dimensolbipentexdex = s(3,3{1{1{3}2}1,2}2) = s(3,3{1{1{3}2}3}2), a.k.a. dimensolplexbipentex.
  • dimensoldexbipentex = s(3,3{1{1{3}2}1{1{3}2}2}2)
  • dimensolbipentexthrex = s(3,3{1{2{3}2}2}2)
  • dimensolbipentextetrex = s(3,3{1{1{3}1,2}2}2) = s(3,3{1{1{3}3}2}2), a.k.a. dimensolthrexbipentex.
  • dimensoltetrexbipentex = s(3,3{1{1{3}1{3}2}2}2)
  • dimensoltripentex = s(3,3{1{1{4}2}2}2)
  • dimensolquadripentex = s(3,3{1{1{5}2}2}2)
  • dimensolquintipentex = s(3,3{1{1{6}2}2}2)

Using suffix “-hex”, we have a naming space of {\omega^{\omega5}}.

  • dimentrienhex = s(3,3,3{1{1{1,2}2}2}2)
  • dimentrilhex = s(3,3{1{1{1,2}2}2}3)
  • dimenbolhex = s(3,3{1{1{1,2}2}2}1{1{1{1,2}2}2}2)
  • dimensolhexplex = s(3,3{2{1{1,2}2}2}2)
  • dimensolhexdex = s(3,3{1{1{1,2}2}1,2}2) = s(3,3{1{1{1,2}2}3}2), a.k.a. dimensolplexhex.
  • dimensoldexhex = s(3,3{1{1{1,2}2}1{1{1,2}2}2}2)
  • dimensolhexthrex = s(3,3{1{2{1,2}2}2}2)
  • dimensolhextetrex = s(3,3{1{1{1,2}1,2}2}2) = s(3,3{1{1{1,2}3}2}2), a.k.a. dimensolthrexhex.
  • dimensoltetrexhex = s(3,3{1{1{1,2}1{1,2}2}2}2)
  • dimensolhexpentex = s(3,3{1{1{2,2}2}2}2)
  • dimensolbihex = s(3,3{1{1{1,1,2}2}2}2) = s(3,3{1{1{1,3}2}2}2), a.k.a. dimensolpentexhex.
  • dimensolheptex = s(3,3{1{1{1{2}2}2}2}2) = s(3,3{1{1{1,1,1,2}2}2}2), a.k.a. dimensoltrihex.
  • dimensolquadrihex = s(3,3{1{1{1,1,1,1,2}2}2}2)
  • dimensolquintihex = s(3,3{1{1{1,1,1,1,1,2}2}2}2)

And further…

  • dimentrienheptex = s(3,3,3{1{1{1{2}2}2}2}2)
  • dimentrilheptex = s(3,3{1{1{1{2}2}2}2}3)
  • dimenbolheptex = s(3,3{1{1{1{2}2}2}2}1{1{1{1{2}2}2}2}2)
  • dimensoloctex = s(3,3{1{1{1{1,2}2}2}2}2) = s(3,3{1{1{1{3}2}2}2}2), a.k.a. dimensolbiheptex.
  • dimensolbioctex = s(3,3{1{1{1{1,1,2}2}2}2}2)
  • dimensolennex = s(3,3{1{1{1{1{2}2}2}2}2}2) = s(3,3{1{1{1{1,1,1,2}2}2}2}2), a.k.a. dimensoltrioctex.
  • dimensoldecex = s(3,3{1{1{1{1{1,2}2}2}2}2}2) = s(3,3{1{1{1{1{3}2}2}2}2}2), a.k.a. dimensolbiennex.

So an “n-ex” means {\omega^{\omega^{.^{.^{.^{\omega^\alpha}}}}}\mapsto\omega^{\omega^{.^{.^{.^{\omega^{\alpha+\beta}}}}}}} (where the α is at the n-th level of the power tower), where β is defined by the result of adding k suffixes in previous steps, where k is the position of the suffix we focus from the end of the name. Extending the suffixes to “n-ex” for any n, we have a naming space of {\omega^{\omega^2}}, which is certainly not enough for every separator recursion level in exAN. But we can add modifiers “n-ex-r” (where r is in Roman numerals), then we have a naming space of {\varepsilon_0} – that’s enough for every separator recursion level in exAN.

What’s more, ε0 is the first ordinal that separator recursion level catches up with growth rate (in FGH scale). If we build a “junior” naming system, where the only difference from the normal one is that the “n-ex” rule is for growth rate instead of separator recursion level, then the “n-ex” is analogous to the “(n+2)-ex” in the “junior” naming system. e.g.

dimensolennex jr. = s(3,3{1{1{1{2}2}2}2}2) = dimensolheptex

So we can name every growth rate if we use the “junior” naming system. But here I don’t use that because I don’t want to left “-plex” and “-dex” below separators (which happens in the “junior” naming system), and separators are important in parts beyond exAN.