Ordinal notations (part 3) – Taranovsky’s notation

Taranovsky’s notation is a simple yet strong ordinal notation, introduced by Dmytro Taranovsky.

Definition

First, Taranovsky’s notation is actually made up of many systems, called 1st system, 2nd system, 3rd system, and so on.

In the n-th system, we uses a binary function: C(α,β), and two constants: 0 and Ωn. Here the 0 is not an ordinal or a number – it’s just a notation for “something”. And the same, Ωn is not an ordinal – it’s just “something”.

Here’s the definition of standard form. First, 0 and Ωn are in standard form. Then C(α,β) is in standard form iff it fits all those shown below:

  1. α and β are in standard form
  2. β = 0, or β = Ωn, or β = C(γ,δ) with α ≤ γ
  3. α is n-built from below from <C(α,β)

But what’s “≤” and what’s “n-built from below from”? To answer this question, we need to define some more things.

First we need to define “n-built from below from” as follows.

  • α is 0-built from below from <β iff α < β.
  • α is (k+1)-built from below from β iff for all subterm γ of α, γ ≤ α or there is such a subterm δ of α that γ is subterm of δ and δ is k-built from below from β.
  • The “subterm” in α can be defined as follows.
    • In any part of expression of α, η is subterm of η itself.
    • In any part of expression of α, if η = C(γ,δ), and x is a subterm of γ or δ, then x is a subterm of η.

To define binary relation “<” and “≤”, we need another form of the notation – postfix form. We can make a postfix form of every normal expression in such a way. First delete all the (′s, )′s and commas, which means we’ll get a string only containing 0, Ωn and C’s. Then reverse the string. Now we get the postfix form. For expressions in standard form, we make the postfix form of them, then the comparisons (“≤”, “≥”, “<” and “>”) are done in the lexicographical order where C < 0 < Ωn.

Property: In n-th system, for b < Ωn, Ωn is 1-built from below from b but not 0-built from below from b.

So why is it an ordinal notation? Because the “<” relation of expressions in standard form fits the ordinal axioms.

Now, we know that 0 is truly the ordinal 0. But unlike in θ function, Ωn is not the n-th uncountable cardinal – it’s just some very large ordinal, but they’re also such common fixed point that {\omega^{\Omega_n}=\Omega_n,\;\varphi(\Omega_n,0)=\Omega_n,\;\Gamma_{\Omega_n}=\Omega_n}, etc.

One variable function

Any ordinal α can be expressed as {\alpha=C(\beta_1,C(\beta_2,\cdots C(\beta_k,0)))} or {\alpha=C(\beta_1,C(\beta_2,\cdots C(\beta_k,\Omega_n)))} in standard form (k can also be 0). The former one is less than Ωn (we call it low-ordinal here) while the latter one is greaterequal to Ωn (we call it high-ordinal here).

Define the one variable function C1 as follows, it’s actually another representation of ordinals: for low-ordinal α, {\alpha=C(\beta_1,C(\beta_2,\cdots C(\beta_{k-1},C(\beta_k,0))))=C_1(\omega^{\beta_k}+\omega^{\beta_{k-1}}+\cdots+\omega^{\beta_2}+\omega^{\beta_1})}.

For high-ordinal α, {\alpha=C(\beta_1,C(\beta_2,\cdots C(\beta_{k-1},C(\beta_k,\Omega_n))))=\Omega_n+\omega^{\beta_k}+\omega^{\beta_{k-1}}+\cdots+\omega^{\beta_2}+\omega^{\beta_1}}, which is the same as ordinal addition and exponentiation.

It turns out that, if {\alpha<\varepsilon_0}, then C1(α) = α, and C1(β+α) = C1(β)+α.

All the 1st system, 2nd system, 3rd system, n-th systems can be combined into one notation as follows: the constants are 0 and Ωn (for every positive integer n), and binary function C. Ωi = C(Ωi+1, 0) and the standard form always uses Ωi instead of C(Ωi+1, 0). And, to check for standard form and compare ordinals, use Ωi = C(Ωi+1, 0) to convert each Ω to Ωn for a single positive integer n (it does not matter which n) and then use the n-th ordinal notation system.

Here‘s a calculator in Python program, which convert ordinals into Cantor’s normal form, binary φ function, and C1-representation. Just note that it uses “C” instead of “C1” for the one variable function, and “C2” instead of “C” for the binary function.

Analysis

Below {\varepsilon_0}, doesn’t appear. For expressions without any Ωn′s, α is 1-built from below from β, so “α is n-built from below from β” is always true.

So C(0,0) = 1, C(0,C(0,0)) = 2, C(0,C(0,C(0,0))) = 3, and C(0,α) = α+1.
Then {C(C(0,0),0)=C(1,0)=\omega} and {C(1,\alpha)=\alpha+\omega}.
Then {C(C(0,C(0,0)),0)=C(2,0)=\omega^2} and {C(2,\alpha)=\alpha+\omega^2}.
{C(C(C(0,0),0),0)=C(\omega,0)=\omega^\omega} and {C(\omega,\alpha)=\alpha+\omega^\omega}.
{C(C(0,C(C(0,0),0)),0)=C(\omega+1,0)=\omega^{\omega+1}} and {C(\omega+1,\alpha)=\alpha+\omega^{\omega+1}}.
{C(C(C(0,0),C(C(0,0),0)),0)=C(\omega2,0)=\omega^{\omega2}} and {C(\omega2,\alpha)=\alpha+\omega^{\omega2}}.
{C(C(C(0,C(0,0)),0),0)=C(\omega^2,0)=\omega^{\omega^2}} and {C(\omega^2,\alpha)=\alpha+\omega^{\omega^2}}.
{C(C(C(C(0,0),0),0),0)=C(\omega^\omega,0)=\omega^{\omega^\omega}} and {C(\omega^\omega,\alpha)=\alpha+\omega^{\omega^\omega}}.
It turns out that, for {\alpha<\Omega_1,\;C(\alpha,\beta)=\beta+\omega^\alpha}, if C(α,β) is standard.

Here’re more results in 1st system.

{C(\Omega_1,0)=C_1(\Omega_1)=\varepsilon_0}
{C(\varepsilon_0,\varepsilon_0)=\varepsilon_02}
{C(\varepsilon_0,C(\varepsilon_0,\varepsilon_0))=\varepsilon_03}
{C(\varepsilon_0+1,\varepsilon_0)=\omega^{\varepsilon_0+1}}
{C(\varepsilon_02,\varepsilon_0)=\omega^{\varepsilon_02}}
{C(\omega^{\varepsilon_0+1},\varepsilon_0)=\omega^{\omega^{\varepsilon_0+1}}}
{C(\omega^{\varepsilon_02},\varepsilon_0)=\omega^{\omega^{\varepsilon_02}}}
{C(\omega^{\omega^{\varepsilon_0+1}},\varepsilon_0)=\omega^{\omega^{\omega^{\varepsilon_0+1}}}}
{C(\Omega_1,C(\Omega_1,0))=C_1(\Omega_12)=\varepsilon_1}
{C(\Omega_1,C(\Omega_1,C(\Omega_1,0)))=C_1(\Omega_13)=\varepsilon_2}
{C(C(0,\Omega_1),0)=C(\Omega_1+1,0)=C_1(\Omega_1\omega)=\varepsilon_\omega}. Note that ordinals above Ω1 are high-ordinals and C(0,Ω1) = Ω1+1 in 1st system.
{C(\Omega_1,C(\Omega_1+1,0))=C_1(\Omega_1\omega+\Omega_1)=\varepsilon_{\omega+1}}
{C(\Omega_1+1,C(\Omega_1+1,0))=C_1(\Omega_1\omega2)=\varepsilon_{\omega2}}
{C(\Omega_1+2,0)=C_1(\Omega_1\omega^2)=\varepsilon_{\omega^2}}
{C(\Omega_1+3,0)=C_1(\Omega_1\omega^3)=\varepsilon_{\omega^3}}
{C(\Omega_1+\omega,0)=C_1(\Omega_1\omega^\omega)=\varepsilon_{\omega^\omega}}
{C(\Omega_1+\omega+1,0)=C_1(\Omega_1\omega^{\omega+1})=\varepsilon_{\omega^{\omega+1}}}
{C(\Omega_1+\omega^2,0)=C_1(\Omega_1\omega^{\omega^2})=\varepsilon_{\omega^{\omega^2}}}
{C(\Omega_1+\omega^\omega,0)=C_1(\Omega_1\omega^{\omega^\omega})=\varepsilon_{\omega^{\omega^\omega}}}
{C(\Omega_1+\varepsilon_0,0)=C_1(\Omega_1\varepsilon_0)=\varepsilon_{\varepsilon_0}}
{C(\Omega_1+\varepsilon_{\varepsilon_0},0)=C_1(\Omega_1\varepsilon_{\varepsilon_0})=\varepsilon_{\varepsilon_{\varepsilon_0}}}
{C(\Omega_12,0)=C_1(\Omega_1^2)=\varphi(2,0)}
{C(\Omega_1,\varphi(2,0))=C_1(\Omega_1^2+\Omega_1)=\varepsilon_{\varphi(2,0)+1}}
{C(\Omega_1+1,\varphi(2,0))=\varepsilon_{\varphi(2,0)+\omega}}
{C(\Omega_1+\varepsilon_0,\varphi(2,0))=\varepsilon_{\varphi(2,0)+\varepsilon_0}}
{C(\Omega_1+\varepsilon_{\varepsilon_0},\varphi(2,0))=\varepsilon_{\varphi(2,0)+\varepsilon_{\varepsilon_0}}}
{C(\Omega_1+\varphi(2,0),\varphi(2,0))=\varepsilon_{\varphi(2,0)2}}
{C(\Omega_1+\varphi(2,0)+1,\varphi(2,0))=\varepsilon_{\omega^{\varphi(2,0)+1}}}
{C(\Omega_1+\varphi(2,0)2,\varphi(2,0))=\varepsilon_{\omega^{\varphi(2,0)2}}}
{C(\Omega_1+C(\Omega_1,\varphi(2,0)),\varphi(2,0))=\varepsilon_{\varepsilon_{\varphi(2,0)+1}}}
{C(\Omega_1+C(\Omega_1+C(\Omega_1,\varphi(2,0)),\varphi(2,0)),\varphi(2,0))=\varepsilon_{\varepsilon_{\varepsilon_{\varphi(2,0)+1}}}}
{C(\Omega_12,C(\Omega_12,0))=C_1(\Omega_1^22)=\varphi(2,1)}
{C(\Omega_12+1,0)=\varphi(2,\omega)}
{C(\Omega_12+C(\Omega_12,0),0)=\varphi(2,\varphi(2,0))}
{C(\Omega_13,0)=C_1(\Omega_1^3)=\varphi(3,0)}
{C(\Omega_1\omega,0)=C_1(\Omega_1^\omega)=\varphi(\omega,0)}
{C(\Omega_1,C(\Omega_1\omega,0))=C_1(\Omega_1^\omega+\Omega_1)=\varepsilon_{\varphi(\omega,0)+1}}
{C(\Omega_12,C(\Omega_1\omega,0))=C_1(\Omega_1^\omega+\Omega_1^2)=\varphi(2,\varphi(\omega,0)+1)}
{C(\Omega_1\omega,C(\Omega_1\omega,0))=C_1(\Omega_1^\omega2)=\varphi(\omega,1)}
{C(\Omega_1\omega+1,0)=\varphi(\omega,\omega)}
{C(\Omega_1\omega+\Omega_1,0)=C_1(\Omega_1^{\omega+1})=\varphi(\omega+1,0)}
{C(\Omega_1\omega2,0)=C_1(\Omega_1^{\omega2})=\varphi(\omega2,0)}
{C(\Omega_1\omega^2,0)=C_1(\Omega_1^{\omega^2})=\varphi(\omega^2,0)}
{C(\Omega_1C(\Omega_1,0),0)=\varphi(\varepsilon_0,0)}
{C(\Omega_1C(\Omega_12,0),0)=\varphi(\varphi(2,0),0)}
{C(\Omega_1C(\Omega_1\omega,0),0)=\varphi(\varphi(\omega,0),0)}
{C(\Omega_1C(\Omega_1C(\Omega_1,0),0),0)=\varphi(\varphi(\varepsilon_0,0),0)}
{C(\Omega_1^2,0)=C_1(\Omega_1^{\Omega_1})=\Gamma_0}
{C(\Omega_1,\Gamma_0)=\varepsilon_{\Gamma_0+1}}
{C(\Omega_1+\Gamma_0,\Gamma_0)=\varepsilon_{\Gamma_02}}
{C(\Omega_12,\Gamma_0)=\varphi(2,\Gamma_0+1)}
{C(\Omega_1\Gamma_0,\Gamma_0)=\varphi(\Gamma_0,1)}
{C(\Omega_1C(\Omega_1\Gamma_0,\Gamma_0),\Gamma_0)=\varphi(\varphi(\Gamma_0,1),0)}
{C(\Omega_1^2,C(\Omega_1^2,0))=C_1(\Omega_1^{\Omega_1}2)=\Gamma_1}
{C(\Omega_1^2+1,0)=\Gamma_\omega}
{C(\Omega_1^2+\Omega_1,0)=\theta(\Omega+1,0)}
{C(\Omega_1^2+\Omega_12,0)=\theta(\Omega+2,0)}
{C(\Omega_1^2+\Omega_1C(\Omega_1^2,0),0)=\theta(\Omega+\theta(\Omega,0),0)}
{C(\Omega_1^2+\Omega_1C(\Omega_1^2+1,0),0)=\theta(\Omega+\theta(\Omega,\omega),0)}
{C(\Omega_1^2+\Omega_1C(\Omega_1^2+\Omega_1C(\Omega_1^2,0),0),0)=\theta(\Omega+\theta(\Omega+\theta(\Omega,0),0),0)}
{C(\Omega_1^22,0)=C_1(\Omega_1^{\Omega_12})=\theta(\Omega2,0)}
{C(\Omega_1^23,0)=\theta(\Omega3,0)}
{C(\Omega_1^2\omega,0)=\theta(\Omega\omega,0)}
{C(\Omega_1^2C(\Omega_1^2,0),0)=\theta(\Omega\theta(\Omega,0),0)}
{C(\Omega_1^2C(\Omega_1^2C(\Omega_1^2,0),0),0)=\theta(\Omega\theta(\Omega\theta(\Omega,0),0),0)}
{C(\Omega_1^3,0)=C_1(\Omega_1^{\Omega_1^2})=\theta(\Omega^2,0)}
{C(\Omega_1^4,0)=C_1(\Omega_1^{\Omega_1^3})=\theta(\Omega^3,0)}
{C(\Omega_1^\omega,0)=C_1(\Omega_1^{\Omega_1^\omega})=\theta(\Omega^\omega,0)}
{C(\Omega_1^\omega,C(\Omega_1^\omega,0))=\theta(\Omega^\omega,1)}
{C(\Omega_1^\omega+1,0)=\theta(\Omega^\omega,\omega)}
{C(\Omega_1^\omega+\Omega_1,0)=\theta(\Omega^\omega+1,0)}
{C(\Omega_1^\omega+\Omega_12,0)=\theta(\Omega^\omega+2,0)}
{C(\Omega_1^\omega+\Omega_1^2,0)=\theta(\Omega^\omega+\Omega,0)}
{C(\Omega_1^\omega+\Omega_1^3,0)=\theta(\Omega^\omega+\Omega^2,0)}
{C(\Omega_1^\omega2,0)=\theta(\Omega^\omega2,0)}
{C(\Omega_1^{\omega+1},0)=\theta(\Omega^{\omega+1},0)}
{C(\Omega_1^{C(\Omega_1^\omega,0)},0)=\theta(\Omega^{\theta(\Omega^\omega,0)},0)}
{C(\Omega_1^{\Omega_1},0)=\theta(\Omega^{\Omega},0)}
{C(\Omega_1^{C(\Omega_1^{\Omega_1},0)},C(\Omega_1^{\Omega_1},0))=\theta(\Omega^{\theta(\Omega^\Omega,0)},1)}
{C(\Omega_1^{\Omega_1},C(\Omega_1^{\Omega_1},0))=\theta(\Omega^\Omega,1)}
{C(\Omega_1^{\Omega_1}+1,0)=\theta(\Omega^\Omega,\omega)}
{C(\Omega_1^{\Omega_1}+\Omega_1,0)=\theta(\Omega^\Omega+1,0)}
{C(\Omega_1^{\Omega_1}2,0)=\theta(\Omega^\Omega2,0)}
{C(\Omega_1^{\Omega_1+1},0)=\theta(\Omega^{\Omega+1},0)}
{C(\Omega_1^{\Omega_12},0)=\theta(\Omega^{\Omega2},0)}
{C(\Omega_1^{\Omega_1^2},0)=\theta(\Omega^{\Omega^2},0)}
{C(\Omega_1^{\Omega_1^{\Omega_1}},0)=\theta(\Omega^{\Omega^\Omega},0)}
{C(\Omega_1^{\Omega_1^{\Omega_1^{\Omega_1}}},0)=\theta(\Omega^{\Omega^{\Omega^\Omega}},0)}

It seems that the Ω1 in C function works very similar to the Ω in θ function. And they are also very similar to my EAN: Ω1 or Ω expands into “ω layers of” the C or θ where they are, and the results are the supremum of any finite layers of the C or θ where the Ω1 or Ω is.

The 1st system only handles ordinals up to the Bachmann-Howard ordinal. So we need 2nd system now. In 2nd system, Ω1 = C(Ω2,0), and the Bachmann-Howard ordinal is the supremum of all the C(C(Ω2,0),0), C(C(C(Ω2,0),C(Ω2,0)),0), C(C(C(C(Ω2,0),C(Ω2,0)),C(Ω2,0)),0), C(C(C(C(C(Ω2,0),C(Ω2,0)),C(Ω2,0)),C(Ω2,0)),0), etc. Thus the BHO equals {C(C(\Omega_2,C(\Omega_2,0)),0)=C(C(\Omega_2,\Omega_1),0)}.

{C(\Omega_1,C(C(\Omega_2,\Omega_1),0))=\varepsilon_{\theta(\varepsilon_{\Omega+1},0)+1}}
{C(\Omega_1^{\Omega_1},C(C(\Omega_2,\Omega_1),0))=\theta(\Omega^\Omega,\theta(\varepsilon_{\Omega+1},0)+1)}
{C(C(\Omega_2,\Omega_1),C(C(\Omega_2,\Omega_1),0))=\theta(\varepsilon_{\Omega+1},1)}
{C(C(\Omega_2,\Omega_1)+1,0)=\theta(\varepsilon_{\Omega+1},\omega)}
{C(C(\Omega_2,\Omega_1)+\Omega_1,0)=\theta(\varepsilon_{\Omega+1}+1,0)}
{C(C(\Omega_2,\Omega_1)+\Omega_1^2,0)=\theta(\varepsilon_{\Omega+1}+\Omega,0)}
{C(C(\Omega_2,\Omega_1)+\Omega_1^{\Omega_1},0)=\theta(\varepsilon_{\Omega+1}+\Omega^\Omega,0)}
{C(C(\Omega_2,\Omega_1)+C(C(\Omega_2,\Omega_1),\Omega_1),0)=\theta(\varepsilon_{\Omega+1}2,0)}. This ordinal (C(C(C(C(Ω21),Ω1),C(Ω21)),0)) is the supremum of all the C(C(C(Ω11),C(Ω21)),0), C(C(C(C(Ω11),Ω1),C(Ω21)),0), C(C(C(C(C(Ω11),Ω1),Ω1),C(Ω21)),0), etc. It’s not C(C(Ω21)2,0)=C(C(C(Ω2,C(Ω2,0)),C(Ω2,C(Ω2,0))),0)! And {C(C(\Omega_2,\Omega_1),\Omega_1)=\varepsilon_{\Omega_1+1}} is standard in 2nd system.
{C(C(\Omega_2,\Omega_1)+C(C(\Omega_2,\Omega_1),\Omega_1)+\Omega_1,0)=\theta(\varepsilon_{\Omega+1}2+1,0)}
{C(C(\Omega_2,\Omega_1)+C(C(\Omega_2,\Omega_1),\Omega_1)2,0)=\theta(\varepsilon_{\Omega+1}3,0)}
{C(C(\Omega_2,\Omega_1)+\omega^{C(C(\Omega_2,\Omega_1),\Omega_1)+1},0)=\theta(\omega^{\varepsilon_{\Omega+1}+1},0)}
{C(C(\Omega_2,\Omega_1)+\omega^{C(C(\Omega_2,\Omega_1),\Omega_1)2},0)=\theta(\omega^{\varepsilon_{\Omega+1}2},0)}
{C(C(\Omega_2,\Omega_1)+\omega^{\omega^{C(C(\Omega_2,\Omega_1),\Omega_1)+1}},0)=\theta(\omega^{\omega^{\varepsilon_{\Omega+1}+1}},0)}
{C(C(\Omega_2,\Omega_1)+C(C(\Omega_2,\Omega_1),C(C(\Omega_2,\Omega_1),\Omega_1)),0)=\theta(\varepsilon_{\Omega+2},0)}
{C(C(\Omega_2,\Omega_1)+C(C(\Omega_2,\Omega_1)+1,\Omega_1),0)=\theta(\varepsilon_{\Omega+\omega},0)}. Note again that C(0,C(Ω21)) = C(Ω21)+1.
{C(C(\Omega_2,\Omega_1)+C(C(\Omega_2,\Omega_1)+\Omega_1,\Omega_1),0)=\theta(\varepsilon_{\Omega2},0)}
{C(C(\Omega_2,\Omega_1)+C(C(\Omega_2,\Omega_1)+\Omega_1^{\Omega_1},\Omega_1),0)=\theta(\varepsilon_{\Omega^\Omega},0)}
{C(C(\Omega_2,\Omega_1)+C(C(\Omega_2,\Omega_1)+C(C(\Omega_2,\Omega_1),\Omega_1),\Omega_1),0)=\theta(\varepsilon_{\varepsilon_{\Omega+1}},0)}
{C(C(\Omega_2,\Omega_1)+C(C(\Omega_2,\Omega_1)+C(C(\Omega_2,\Omega_1)+1,\Omega_1),\Omega_1),0)=\theta(\varepsilon_{\varepsilon_{\Omega+\omega}},0)}
{C(C(\Omega_2,\Omega_1)+C(C(\Omega_2,\Omega_1)+C(C(\Omega_2,\Omega_1)+C(C(\Omega_2,\Omega_1),\Omega_1),\Omega_1),\Omega_1),0)}\\ {=\theta(\varepsilon_{\varepsilon_{\varepsilon_{\Omega+1}}},0)}
{C(C(\Omega_2,\Omega_1)2,0)=\theta(\varphi(2,\Omega+1),0)}. This ordinal (C(C(C(Ω21),C(Ω21)),0)) is the supremum of all the C(C(Ω21),0), C(C(C(C(Ω21),Ω1),C(Ω21)),0), C(C(C(C(C(C(Ω21),Ω1),C(Ω21)),Ω1),C(Ω21)),0), etc.
{C(C(\Omega_2,\Omega_1),C(C(\Omega_2,\Omega_1)2,0))=\theta(\varepsilon_{\Omega+1},\theta(\varphi(2,\Omega+1),0)+1)}
{C(C(\Omega_2,\Omega_1)+C(C(\Omega_2,\Omega_1),\Omega_1),C(C(\Omega_2,\Omega_1)2,0))=\theta(\varepsilon_{\Omega+1}2,\theta(\varphi(2,\Omega+1),0)+1)}
{C(C(\Omega_2,\Omega_1)+C(C(\Omega_2,\Omega_1)+C(C(\Omega_2,\Omega_1),\Omega_1),\Omega_1),C(C(\Omega_2,\Omega_1)2,0))}\\ {=\theta(\varepsilon_{\varepsilon_{\Omega+1}},\theta(\varphi(2,\Omega+1),0)+1)}
{C(C(\Omega_2,\Omega_1)2,C(C(\Omega_2,\Omega_1)2,0))=\theta(\varphi(2,\Omega+1),1)}
{C(C(\Omega_2,\Omega_1)2+1,0)=\theta(\varphi(2,\Omega+1),\omega)}
{C(C(\Omega_2,\Omega_1)2+\Omega_1,0)=\theta(\varphi(2,\Omega+1)+1,0)}
{C(C(\Omega_2,\Omega_1)2+C(C(\Omega_2,\Omega_1),\Omega_1),0)=\theta(\varphi(2,\Omega+1)+\varepsilon_{\Omega+1},0)}
{C(C(\Omega_2,\Omega_1)2+C(C(\Omega_2,\Omega_1)+C(C(\Omega_2,\Omega_1),\Omega_1),\Omega_1),0)}\\ {=\theta(\varphi(2,\Omega+1)+\varepsilon_{\varepsilon_{\Omega+1}},0)}
{C(C(\Omega_2,\Omega_1)2+C(C(\Omega_2,\Omega_1)2,\Omega_1),0)=\theta(\varphi(2,\Omega+1)2,0)}.
{C(C(\Omega_2,\Omega_1)2+C(C(\Omega_2,\Omega_1)2,\Omega_1)2,0)=\theta(\varphi(2,\Omega+1)3,0)}
{C(C(\Omega_2,\Omega_1)2+C(C(\Omega_2,\Omega_1),C(C(\Omega_2,\Omega_1)2,\Omega_1)),0)=\theta(\varepsilon_{\varphi(2,\Omega+1)+1},0)}
{C(C(\Omega_2,\Omega_1)2+C(C(\Omega_2,\Omega_1)+C(C(\Omega_2,\Omega_1)2,\Omega_1),C(C(\Omega_2,\Omega_1)2,\Omega_1)),0)}\\ {=\theta(\varepsilon_{\varphi(2,\Omega+1)2},0)}
{C(C(\Omega_2,\Omega_1)2+C(C(\Omega_2,\Omega_1)2,C(C(\Omega_2,\Omega_1)2,\Omega_1)),0)=\theta(\varphi(2,\Omega+2),0)}
{C(C(\Omega_2,\Omega_1)2+C(C(\Omega_2,\Omega_1)2+1,\Omega_1),0)=\theta(\varphi(2,\Omega+\omega),0)}
{C(C(\Omega_2,\Omega_1)2+C(C(\Omega_2,\Omega_1)2+\Omega_1,\Omega_1),0)=\theta(\varphi(2,\Omega2),0)}
{C(C(\Omega_2,\Omega_1)2+C(C(\Omega_2,\Omega_1)2+C(C(\Omega_2,\Omega_1)2,\Omega_1),\Omega_1),0)}\\ {=\theta(\varphi(2,\varphi(2,\Omega+1)),0)}
{C(C(\Omega_2,\Omega_1)3,0)=\theta(\varphi(3,\Omega+1),0)}
{C(\omega^{C(\Omega_2,\Omega_1)+1},0)=\theta(\varphi(\omega,\Omega+1),0)}
{C(\omega^{C(\Omega_2,\Omega_1)+1}+C(\Omega_2,\Omega_1),0)=\theta(\varphi(\omega+1,\Omega+1),0)}
{C(\omega^{C(\Omega_2,\Omega_1)+2},0)=\theta(\theta(\omega^2,\Omega),0)}
{C(\omega^{C(\Omega_2,\Omega_1)+\Omega_1},0)=\theta(\theta(\Omega,\Omega),0)}
{C(\omega^{C(\Omega_2,\Omega_1)+C(C(\Omega_2,\Omega_1),\Omega_1)},0)=\theta(\theta(\varepsilon_{\Omega+1},\Omega),0)}
{C(\omega^{C(\Omega_2,\Omega_1)+C(C(\Omega_2,\Omega_1)+1,\Omega_1)},0)=\theta(\theta(\varepsilon_{\Omega+\omega},\Omega),0)}
{C(\omega^{C(\Omega_2,\Omega_1)+C(C(\Omega_2,\Omega_1)+C(C(\Omega_2,\Omega_1),\Omega_1),\Omega_1)},0)=\theta(\theta(\varepsilon_{\varepsilon_{\Omega+1}},\Omega),0)}
{C(\omega^{C(\Omega_2,\Omega_1)+C(C(\Omega_2,\Omega_1)2,\Omega_1)},0)=\theta(\theta(\theta(2,\Omega),\Omega),0)}
{C(\omega^{C(\Omega_2,\Omega_1)+C(\omega^{C(\Omega_2,\Omega_1)+1},\Omega_1)},0)=\theta(\theta(\theta(\omega,\Omega),\Omega),0)}
{C(\omega^{C(\Omega_2,\Omega_1)+C(\omega^{C(\Omega_2,\Omega_1)+\Omega_1},\Omega_1)},0)=\theta(\theta(\theta(\Omega,\Omega),\Omega),0)}
{C(\omega^{C(\Omega_2,\Omega_1)2},0)=\theta(\Omega_2,0)}. This ordinal (C(C(C(C(Ω21),C(Ω21)),C(Ω21)),0)) is the supremum of all the C(C(C(Ω21),C(Ω21)),0), C(C(C(C(C(C(Ω21),C(Ω21)),Ω1),C(Ω21)),C(Ω21)),0), C(C(C(C(C(C(C(C(C(Ω21),C(Ω21)),Ω1),C(Ω21)),C(Ω21)),Ω1),C(Ω21)),C(Ω21)),0), etc.
{C(\omega^{C(\Omega_2,\Omega_1)2}+1,0)=\theta(\Omega_2,\omega)}
{C(\omega^{C(\Omega_2,\Omega_1)2}+\Omega_1,0)=\theta(\Omega_2+1,0)}
{C(\omega^{C(\Omega_2,\Omega_1)2}+C(C(\Omega_2,\Omega_1),\Omega_1),0)=\theta(\Omega_2+\varepsilon_{\Omega+1},0)}
{C(\omega^{C(\Omega_2,\Omega_1)2}+C(C(\Omega_2,\Omega_1)2,\Omega_1),0)=\theta(\Omega_2+\theta(2,\Omega),0)}
{C(\omega^{C(\Omega_2,\Omega_1)2}+C(\omega^{C(\Omega_2,\Omega_1)+\Omega_1},\Omega_1),0)=\theta(\Omega_2+\theta(\Omega,\Omega),0)}
{C(\omega^{C(\Omega_2,\Omega_1)2}+C(\omega^{C(\Omega_2,\Omega_1)+C(C(\Omega_2,\Omega_1),\Omega_1)},\Omega_1),0)=\theta(\Omega_2+\theta(\varepsilon_{\Omega+1},\Omega),0)}
{C(\omega^{C(\Omega_2,\Omega_1)2}+C(\omega^{C(\Omega_2,\Omega_1)2},\Omega_1),0)=\theta(\Omega_2+\theta(\Omega_2,\Omega),0)}
{C(\omega^{C(\Omega_2,\Omega_1)2}+C(\omega^{C(\Omega_2,\Omega_1)2},\Omega_1)2,0)=\theta(\Omega_2+\theta(\Omega_2,\Omega)2,0)}
{C(\omega^{C(\Omega_2,\Omega_1)2}+C(C(\Omega_2,\Omega_1),C(\omega^{C(\Omega_2,\Omega_1)2},\Omega_1)),0)=\theta(\Omega_2+\varepsilon_{\theta(\Omega_2,\Omega)+1},0)}
{C(\omega^{C(\Omega_2,\Omega_1)2}+C(\omega^{C(\Omega_2,\Omega_1)+\Omega_1},C(\omega^{C(\Omega_2,\Omega_1)2},\Omega_1)),0)=\theta(\Omega_2+\theta(\Omega,\theta(\Omega_2,\Omega)+1),0)}
{C(\omega^{C(\Omega_2,\Omega_1)2}+C(\omega^{C(\Omega_2,\Omega_1)+C(\omega^{C(\Omega_2,\Omega_1)2},\Omega_1)},C(\omega^{C(\Omega_2,\Omega_1)2},\Omega_1)),0)}\\ {=\theta(\Omega_2+\theta(\theta(\Omega_2,\Omega),\Omega+1),0)}
{C(\omega^{C(\Omega_2,\Omega_1)2}+C(\omega^{C(\Omega_2,\Omega_1)2},C(\omega^{C(\Omega_2,\Omega_1)2},\Omega_1)),0)=\theta(\Omega_2+\theta(\Omega_2,\Omega+1),0)}
{C(\omega^{C(\Omega_2,\Omega_1)2}+C(\omega^{C(\Omega_2,\Omega_1)2}+1,\Omega_1),0)=\theta(\Omega_2+\theta(\Omega_2,\Omega+\omega),0)}
{C(\omega^{C(\Omega_2,\Omega_1)2}+C(\omega^{C(\Omega_2,\Omega_1)2}+\Omega_1,\Omega_1),0)=\theta(\Omega_2+\theta(\Omega_2,\Omega2),0)}
{C(\omega^{C(\Omega_2,\Omega_1)2}+C(\omega^{C(\Omega_2,\Omega_1)2}+C(C(\Omega_2,\Omega_1),\Omega_1),\Omega_1),0)=\theta(\Omega_2+\theta(\Omega_2,\varepsilon_{\Omega+1}),0)}
{C(\omega^{C(\Omega_2,\Omega_1)2}+C(\omega^{C(\Omega_2,\Omega_1)2}+C(\omega^{C(\Omega_2,\Omega_1)2},\Omega_1),\Omega_1),0)=\theta(\Omega_2+\theta(\Omega_2,\theta(\Omega_2,\Omega)),0)}
{C(\omega^{C(\Omega_2,\Omega_1)2}+C(\Omega_2,\Omega_1),0)=\theta(\Omega_2+\theta(\Omega_2+1,\Omega),0)}
{C(\omega^{C(\Omega_2,\Omega_1)2}+C(\Omega_2,\Omega_1)2,0)=\theta(\Omega_2+\theta(\Omega_2+2,\Omega),0)}
{C(\omega^{C(\Omega_2,\Omega_1)2}+\omega^{C(\Omega_2,\Omega_1)+\Omega_1},0)=\theta(\Omega_2+\theta(\Omega_2+\Omega,\Omega),0)}
{C(\omega^{C(\Omega_2,\Omega_1)2}+\omega^{C(\Omega_2,\Omega_1)+C(\omega^{C(\Omega_2,\Omega_1)2},\Omega_1)},0)=\theta(\Omega_2+\theta(\Omega_2+\theta(\Omega_2,\Omega),\Omega),0)}
{C(\omega^{C(\Omega_2,\Omega_1)2}+\omega^{C(\Omega_2,\Omega_1)+C(\omega^{C(\Omega_2,\Omega_1)2}+C(\Omega_2,\Omega_1),\Omega_1)},0)}\\ {=\theta(\Omega_2+\theta(\Omega_2+\theta(\Omega_2+1,\Omega),\Omega),0)}
{C(\omega^{C(\Omega_2,\Omega_1)2}+\omega^{C(\Omega_2,\Omega_1)+C(\omega^{C(\Omega_2,\Omega_1)2}+\omega^{C(\Omega_2,\Omega_1)+C(\omega^{C(\Omega_2,\Omega_1)2},\Omega_1)},\Omega_1)},0)}\\ {=\theta(\Omega_2+\theta(\Omega_2+\theta(\Omega_2+\theta(\Omega_2,\Omega),\Omega),\Omega),0)}
{C(\omega^{C(\Omega_2,\Omega_1)2}2,0)=\theta(\Omega_22,0)}
{C(\omega^{C(\Omega_2,\Omega_1)2}3,0)=\theta(\Omega_23,0)}
{C(\omega^{C(\Omega_2,\Omega_1)2+1},0)=\theta(\Omega_2\omega,0)}
{C(\omega^{C(\Omega_2,\Omega_1)2+\Omega_1},0)=\theta(\Omega_2\Omega,0)}
{C(\omega^{C(\Omega_2,\Omega_1)3},0)=\theta(\Omega_2^2,0)}
{C(\omega^{\omega^{C(\Omega_2,\Omega_1)+1}},0)=\theta(\Omega_2^\omega,0)}
{C(\omega^{\omega^{C(\Omega_2,\Omega_1)2}},0)=\theta(\Omega_2^{\Omega_2},0)}
{C(\omega^{\omega^{\omega^{C(\Omega_2,\Omega_1)2}}},0)=\theta(\Omega_2^{\Omega_2^{\Omega_2}},0)}
{C(C(\Omega_2,C(\Omega_2,\Omega_1)),0)=\theta(\varepsilon_{\Omega_2+1},0)}
{C(C(\Omega_2,C(\Omega_2,\Omega_1))+\Omega_1,0)=\theta(\varepsilon_{\Omega_2+1}+1,0)}
{C(C(\Omega_2,C(\Omega_2,\Omega_1))+\omega^{C(\Omega_2,\Omega_1)2},0)=\theta(\varepsilon_{\Omega_2+1}+\Omega_2,0)}
{C(C(\Omega_2,C(\Omega_2,\Omega_1))+\omega^{\omega^{C(\Omega_2,\Omega_1)2}},0)=\theta(\varepsilon_{\Omega_2+1}+\Omega_2^{\Omega_2},0)}
{C(C(\Omega_2,C(\Omega_2,\Omega_1))+C(C(\Omega_2,C(\Omega_2,\Omega_1)),C(\Omega_2,\Omega_1)),0)=\theta(\varepsilon_{\Omega_2+1}2,0)}
{C(C(\Omega_2,C(\Omega_2,\Omega_1))+C(C(\Omega_2,C(\Omega_2,\Omega_1)),C(\Omega_2,\Omega_1))\omega,0)=\theta(\varepsilon_{\Omega_2+1}\omega,0)}
{C(C(\Omega_2,C(\Omega_2,\Omega_1))+C(C(\Omega_2,C(\Omega_2,\Omega_1)),C(C(\Omega_2,C(\Omega_2,\Omega_1)),C(\Omega_2,\Omega_1))),0)}\\ {=\theta(\varepsilon_{\Omega_2+2},0)}
{C(C(\Omega_2,C(\Omega_2,\Omega_1))+C(C(\Omega_2,C(\Omega_2,\Omega_1))+1,C(\Omega_2,\Omega_1)),0)=\theta(\varepsilon_{\Omega_2+\omega},0)}
{C(C(\Omega_2,C(\Omega_2,\Omega_1))+C(C(\Omega_2,C(\Omega_2,\Omega_1))+C(\Omega_2,\Omega_1),C(\Omega_2,\Omega_1)),0)=\theta(\varepsilon_{\Omega_22},0)}
{C(C(\Omega_2,C(\Omega_2,\Omega_1))+C(C(\Omega_2,C(\Omega_2,\Omega_1))+C(C(\Omega_2,C(\Omega_2,\Omega_1)),C(\Omega_2,\Omega_1)),0),C(\Omega_2,\Omega_1)),0)}\\ {=\theta(\varepsilon_{\varepsilon_{\Omega_2+1}},0)}
{C(C(\Omega_2,C(\Omega_2,\Omega_1))2,0)=\theta(\varphi(2,\Omega_2+1),0)}
{C(C(\Omega_2,C(\Omega_2,\Omega_1))3,0)=\theta(\varphi(3,\Omega_2+1),0)}
{C(\omega^{C(\Omega_2,C(\Omega_2,\Omega_1))+1},0)=\theta(\theta(\omega,\Omega_2),0)}
{C(\omega^{C(\Omega_2,C(\Omega_2,\Omega_1))+C(\Omega_2,\Omega_1)},0)=\theta(\theta(\Omega_2,\Omega_2),0)}
{\omega^{C(C(\Omega_2,C(\Omega_2,\Omega_1))+C(C(\Omega_2,C(\Omega_2,\Omega_1)),C(\Omega_2,\Omega_1))},0)=\theta(\theta(\varepsilon_{\Omega_2+1},\Omega_2),0)}
{C(\omega^{C(\Omega_2,C(\Omega_2,\Omega_1))2},0)=\theta(\Omega_3,0)}
{C(\omega^{C(\Omega_2,C(\Omega_2,\Omega_1))2}2,0)=\theta(\Omega_32,0)}
{C(\omega^{C(\Omega_2,C(\Omega_2,\Omega_1))3},0)=\theta(\Omega_3^2,0)}
{C(\omega^{\omega^{C(\Omega_2,C(\Omega_2,\Omega_1))+1}},0)=\theta(\Omega_3^\omega,0)}
{C(\omega^{\omega^{\omega^{C(\Omega_2,C(\Omega_2,\Omega_1))+1}}},0)=\theta(\Omega_3^{\Omega_3^\omega},0)}
{C(C(\Omega_2,C(\Omega_2,C(\Omega_2,\Omega_1))),0)=\theta(\varepsilon_{\Omega_3+1},0)}
{C(C(\Omega_2,C(\Omega_2,C(\Omega_2,C(\Omega_2,\Omega_1)))),0)=\theta(\varepsilon_{\Omega_4+1},0)}
{C(C(\Omega_2+1,0),0)=\theta(\Omega_\omega,0)}
{C(C(\Omega_2+1,0)+\Omega_1,0)=\theta(\Omega_\omega+1,0)}
{C(C(\Omega_2+1,0)+\omega^{C(\Omega_2,\Omega_1)2},0)=\theta(\Omega_\omega+\Omega_2,0)}
{C(C(\Omega_2+1,0)+\omega^{C(\Omega_2,C(\Omega_2,\Omega_1))2},0)=\theta(\Omega_\omega+\Omega_3,0)}
{C(C(\Omega_2+1,0)2,0)=\theta(\Omega_\omega2,0)}
{C(\omega^{C(\Omega_2+1,0)+1},0)=\theta(\Omega_\omega\omega,0)}
{C(\omega^{C(\Omega_2+1,0)2},0)=\theta(\Omega_\omega^2,0)}
{C(\omega^{\omega^{C(\Omega_2+1,0)+1}},0)=\theta(\Omega_\omega^\omega,0)}
{C(\omega^{\omega^{\omega^{C(\Omega_2+1,0)+1}}},0)=\theta(\Omega_\omega^{\Omega_\omega^\omega},0)}
{C(C(\Omega_2,C(\Omega_2+1,0)),0)=\theta(\varepsilon_{\Omega_\omega+1},0)}
{C(C(\Omega_2,C(\Omega_2+1,0))+C(\Omega_2+1,0),0)=\theta(\varepsilon_{\Omega_\omega+1}+\Omega_\omega,0)}
{C(C(\Omega_2,C(\Omega_2+1,0))+\omega^{C(\Omega_2+1,0)2},0)=\theta(\varepsilon_{\Omega_\omega+1}+\Omega_\omega^2,0)}
{C(C(\Omega_2,C(\Omega_2+1,0))+C(C(\Omega_2,C(\Omega_2+1,0)),C(\Omega_2+1,0)),0)=\theta(\varepsilon_{\Omega_\omega+1}2,0)}
{C(C(\Omega_2,C(\Omega_2+1,0))+C(C(\Omega_2,C(\Omega_2+1,0))+1,C(\Omega_2+1,0)),0)=\theta(\varepsilon_{\Omega_\omega+\omega},0)}
{C(C(\Omega_2,C(\Omega_2+1,0))2,0)=\theta(\varphi(2,\Omega_\omega+1),0)}
{C(\omega^{C(\Omega_2,C(\Omega_2+1,0))+1},0)=\theta(\theta(\omega,\Omega_\omega),0)}
{C(\omega^{C(\Omega_2,C(\Omega_2+1,0))+C(\Omega_2+1,0)},0)=\theta(\theta(\Omega_\omega,\Omega_\omega),0)}
{C(\omega^{C(\Omega_2,C(\Omega_2+1,0))+C(C(\Omega_2,C(\Omega_2+1,0)),C(\Omega_2+1,0))},0)=\theta(\theta(\varepsilon_{\Omega_\omega+1},\Omega_\omega),0)}
{C(\omega^{C(\Omega_2,C(\Omega_2+1,0))2},0)=\theta(\Omega_{\omega+1},0)}
{C(C(\Omega_2,C(\Omega_2,C(\Omega_2+1,0))),0)=\theta(\varepsilon_{\Omega_{\omega+1}+1},0)}
{C(C(\Omega_2,C(\Omega_2,C(\Omega_2,C(\Omega_2+1,0)))),0)=\theta(\varepsilon_{\Omega_{\omega+2}+1},0)}
{C(C(\Omega_2+1,C(\Omega_2+1,0)),0)=\theta(\Omega_{\omega2},0)}
{C(C(\Omega_2+1,C(\Omega_2+1,C(\Omega_2+1,0))),0)=\theta(\Omega_{\omega3},0)}
{C(C(\Omega_2+2,0),0)=\theta(\Omega_{\omega^2},0)}
{C(C(\Omega_2+\omega,0),0)=\theta(\Omega_{\omega^\omega},0)}
{C(C(\Omega_2+C(\Omega_1,0),0),0)=\theta(\Omega_{\varepsilon_0},0)}
{C(C(\Omega_2+C(C(\Omega_2,\Omega_1),0),0),0)=\theta(\Omega_{\theta(\varepsilon_{\Omega+1},0)},0)}
{C(C(\Omega_2+C(C(\Omega_2+1,0),0),0),0)=\theta(\Omega_{\theta(\Omega_\omega,0)},0)}
{C(C(\Omega_2+C(C(\Omega_2+C(C(\Omega_2+1,0),0),0),0),0),0)=\theta(\Omega_{\theta(\Omega_{\theta(\Omega_\omega,0)},0)},0)}
{C(C(\Omega_2+\Omega_1,0),0)=\theta(\Omega_\Omega,0)}
{C(C(\Omega_2,C(\Omega_2+\Omega_1,0)),0)=\theta(\varepsilon_{\Omega_\Omega+1},0)}
{C(C(\Omega_2+1,C(\Omega_2+\Omega_1,0)),0)=\theta(\Omega_{\Omega+\omega},0)}
{C(C(\Omega_2+C(C(\Omega_2+\Omega_1,0),0),C(\Omega_2+\Omega_1,0)),0)=\theta(\Omega_{\Omega+\theta(\Omega_\Omega,0)},0)}
{C(C(\Omega_2+\Omega_1,C(\Omega_2+\Omega_1,0)),0)=\theta(\Omega_{\Omega2},0)}
{C(C(\Omega_2+\Omega_1+1,0),0)=\theta(\Omega_{\Omega\omega},0)}
{C(C(\Omega_2+\Omega_12,0),0)=\theta(\Omega_{\Omega^2},0)}
{C(C(\Omega_2+\Omega_1^2,0),0)=\theta(\Omega_{\Omega^\Omega},0)}
{C(C(\Omega_2+\Omega_1^{\Omega_1},0),0)=\theta(\Omega_{\Omega^{\Omega^\Omega}},0)}
{C(C(\Omega_2+C(C(\Omega_2,\Omega_1),\Omega_1),0),0)=\theta(\Omega_{\varepsilon_{\Omega+1}},0)}
{C(C(\Omega_2+C(\omega^{C(\Omega_2,\Omega_1)2},\Omega_1),0),0)=\theta(\Omega_{\theta(\Omega_2,\Omega)},0)}
{C(C(\Omega_2+C(C(\Omega_2+1,0),\Omega_1),0),0)=\theta(\Omega_{\theta(\Omega_\omega,\Omega)},0)}
{C(C(\Omega_2+C(C(\Omega_2+\Omega_1,0),\Omega_1),0),0)=\theta(\Omega_{\theta(\Omega_\Omega,\Omega)},0)}
{C(C(\Omega_2+C(C(\Omega_2+C(C(\Omega_2+1,0),\Omega_1),0),\Omega_1),0),0)=\theta(\Omega_{\theta(\Omega_{\theta(\Omega_\omega,\Omega)},\Omega)},0)}
{C(C(\Omega_2+C(\Omega_2,\Omega_1),0),0)=\theta(\Omega_{\Omega_2},0)}
{C(C(\Omega_2+C(\Omega_2,C(\Omega_2,\Omega_1)),0),0)=\theta(\Omega_{\Omega_3},0)}
{C(C(\Omega_2+C(\Omega_2+1,0),0),0)=\theta(\Omega_{\Omega_\omega},0)}
{C(C(\Omega_2+C(\Omega_2+2,0),0),0)=\theta(\Omega_{\Omega_{\omega^2}},0)}
{C(C(\Omega_2+C(\Omega_2+\Omega_1,0),0),0)=\theta(\Omega_{\Omega_{\Omega}},0)}
{C(C(\Omega_2+C(\Omega_2+C(C(\Omega_2,\Omega_1),\Omega_1),0),0),0)=\theta(\Omega_{\Omega_{\varepsilon_{\Omega+1}}},0)}
{C(C(\Omega_2+C(\Omega_2+C(\Omega_2,\Omega_1),0),0),0)=\theta(\Omega_{\Omega_{\Omega_2}},0)}
{C(C(\Omega_2+C(\Omega_2+C(\Omega_2+1,0),0),0),0)=\theta(\Omega_{\Omega_{\Omega_\omega}},0)}
{C(C(\Omega_2+C(\Omega_2+C(\Omega_2+C(\Omega_2+1,0),0),0),0),0)=\theta(\Omega_{\Omega_{\Omega_{\Omega_\omega}}},0)}

The supremum of all the C(C(C(0,Ω2),0),0), C(C(C(C(C(0,Ω2),0),Ω2),0),0), C(C(C(C(C(C(C(0,Ω2),0),Ω2),0),Ω2),0),0), etc. is C(C(C(Ω22),0),0) = C(C(Ω22,0),0). And that’s the limit of θ function.

The fundamental sequences in Taranovsky’s notation can be simply defined. First, we define a function: L(α), it’s the amount of C’s in the standard form of ordinal α. Then we define:

{\alpha[n]=max\{\beta|\beta<\alpha\land L(\beta)\le L(\alpha)+n\}}

That’s it! So we get a very fast-growing function: {f_{C(C(\cdots C(\Omega_n2,0)\cdots,0),0)}(n)} with n C’s. It marks the strength of Taranovsky’s notation.

The full strength of Taranovsky’s notation is unknown yet. It might be as weak as second-order arithmetic (Z2), and might be as strong as second-order arithmetic with projective determinacy (Z2+PD), according to Taranovsky’s page.

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18 thoughts on “Ordinal notations (part 3) – Taranovsky’s notation

  1. Scorcher says:

    θ(α↦Ω_α) = θ(Ω_Ω_Ω_Ω_Ω_…) ω-times – is limit of θ() function.

    But I can continue:
    θ(Ω_Ω_Ω_Ω_Ω_…) ω-times = α_0
    θ(Ω_Ω_Ω_Ω_Ω_…) α_0-times = α_1
    θ(Ω_Ω_Ω_Ω_Ω_…) α_1-times = α_2
    e.t.c.

    α_ω
    α_θ(Ω)
    α_θ(Ω^Ω)
    α_θ(Ω_Ω)
    α_θ(Ω_Ω_Ω_Ω_Ω_…) = α_α_0
    e.t.c.

    α_α_0
    α_α_α_0
    α_α_α_α_… ω-times = β↦α_β = β_0

    How to write in Taranovsky’s notation or pDAN values:
    α_1
    α_2
    α_ω
    α_α_0
    β_0
    e.t.c.
    ?

    Like

    • Using the 2nd system:

      Your α_0 = C(C(Ω2,0),0)
      α_1 = C(C(Ω2+C(Ω+C(Ω2,0)+α_0,0),0),0)
      α_2 = C(C(Ω2+C(Ω+C(Ω2,0)+α_1,0),0),0)
      α_ω = C(C(Ω2+C(Ω+C(Ω2,0)+C(Ω,0),0),0),0)
      α_α_0 = C(C(Ω2+C(Ω+C(Ω2,0)+C(Ω,0)×α_0,0),0),0)
      β_0 = C(C(Ω2+C(Ω+C(Ω2,0)+C(Ω,0)^2,0),0),0)

      The results seem erratic, because things like C(C(Ω+C(Ω2,0),0),C(C(Ω2,0),0)) aren’t standard.

      Like

      • Scorcher says:

        I’m confused as compared…
        C(C(Ω2+C(Ω+C(Ω2,0)+C(Ω,0),0),0),0) > C(C(Ω2,C(Ω2*2,0)),0) ~ {1,,2{1,,1,,2}2}
        or
        C(C(Ω2+C(Ω+C(Ω2,0)+C(Ω,0),0),0),0) < C(C(Ω2,C(Ω2*2,0)),0) ~ {1,,2{1,,1,,2}2}
        ?

        I just want to compare α_1, α_ω, α_α_0, β_0, etc with pDAN

        Like

        • OK. Compared with pDAN, we have:

          {1,,1{1,,1,,2}2} has recursion level α_0
          {1,,1{1,,1,,2}1{1{1,,1{1,,1,,2}2}2}2} has recursion level α_1
          {1,,1{1,,1,,2}1{1{1,,1{1,,1,,2}1{1{1,,1{1,,1,,2}2}2}2}2}2} has recursion level α_2
          {1,,1{1,,1,,2}1`2} has recursion level α_ω
          {1,,1{1,,1,,2}1`1{1{1,,1{1,,1,,2}2}2}2} has recursion level α_α_0
          {1,,1{1,,1,,2}1`1`2} has recursion level β_0

          Like

          • Scorcher says:

            Do I understand correctly that if I compare
            {1,,1{1,,1,,2}1`2} – ε_0 level of 1`2
            {1,,1{1,,1,,2}1`1`2} – ζ_0 level of 1`1`2
            {1,,1{1,,1,,2}1,,1{1,,1,,2}2} – θ(α↦Ω_α) level of 1,,1{1,,1,,2}2

            сan it be said that the transition from {1,,1{1,,1,,2}2} to {1,,1{2,,1,,2}2} will be:
            ( use some function Ф() )
            α_0 = Ф(ω)
            α_1 = Ф(ω^ω)
            α_2 = Ф(ω^ω^ω)
            α_ω = Ф(φ(1, 0))
            β_0 = Ф(φ(2, 0))
            β_ω = Ф(φ(3, 0))
            e t.c.

            {1,,1{1,,1,,2}1,,1{1,,1,,2}2} – Ф(θ(α↦Ω_α)) = Ф(α) ?

            after:
            Ф(α)
            Ф(Ф(α))
            Ф(Ф(Ф(α)))

            Ф(Ф(Ф(Ф(…)))) – {1,,1{2,,1,,2}2} ?

            Like

    • It’s in lexicographical order. For example, “” < "C" < "CC" < "CCC" < "CCCCCCCCCC" < "C0" < "C0C" < "C0CCCCC" < "C0CC0" < "C0C0" < "C0CΩ" < "C00" < "C0Ω" < "CΩ" < "CΩC" < "CΩ0" < "CΩΩ" < "CΩΩΩ" < "0" < "0C" < "0CC" < "00" < "00C" < "0Ω" < "Ω".
      Generally, the empty string is the least one, "C" is larger, then "0" is larger, then "Ω" is larger. And, if string a < string b, then ca < cb for any string c.

      Like

    • 1 is C(0,0), so your C(1,1) is C(C(0,0),C(0,0)). But it’s not in standard form, because C(0,0) > 0, and it’s against the 2nd rule of standard form.
      In Taranovsky’s notation, we only see expressions in standard form as ordinals.

      Like

.

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