θ function is a binary function. It’s defined as follows:

Where and means the αth uncountable cardinal.

is the first uncountable ordinal and has cardinality , is the smallest ordinal that has cardinality . Generally is the smallest ordinal that has cardinality . They’re so large that , and for all .

It means that, θ(α,β) is the (1+β)-th ordinal such that it cannot be built from ordinals less than it by addition, applying θ(δ,_) where δ < α and getting an uncountable cardinal.

# Explanation

To get we need to deal with C(0,α). C_{0}(0,0) already contains 0, so let’s begin from 1. For C(0,1), C_{0}(0,1) only contains 0, adding and getting uncountable cardinals cannot get 1, so . For C(0,2), 1 is in it, so 2=1+1 is in it. And so on, 3 is in C(0,3), 4 is in C(0,4) ,etc. Until ω, it cannot be built from natural numbers by addition and getting uncountable cardinals, so . Further, .

is the least ordinal that can’t be built from ordinals less than it by addition and applying (and getting uncountable cardinals) – that’s where the Cantor’s normal form cannot get. So . Further, .

It seems that below , making θ function an extension of φ function. Even is true.

Beyond Feferman–Schütte ordinal

However, things suddenly go wrong. In , we cannot use , we can just use for , so .

From on, we can use in , so . And further, .

Then for , and for – It meets another problem at .

, and . Next, , and . Now this pattern where will continue up to Ω, so it’s enough to **define** larger ordinals. But we cannot **name**, say, what’s the first fixed point of , what’s LVO, what’s BHO, etc.

To express large ordinals only using θ function (and Cantor’s normal form), we need to deal with something beyond . Using the result above, for all α < Ω. Now notice that C(Ω+1,α) contains Ω and can apply θ(Ω,α) in it, so is in . In fact, is the (1+β)-th fixed point of .

It seems that, if we write θ function in this way: , where with and , then it’s the (1+β)-th fixed point of where and itself. This pattern holds up to – even beyond Bachmann-Howard ordinal. For example, is the first fixed point of , which is called large Veblen ordinal. The small Veblen ordinal is , and the Bachmann-Howard ordinal is .

Beyond Bachmann-Howard ordinal

The (1+β)-th fixed point of is . For , but – The same thing at happens at Ω now.

Another important point is . What is it? cannot be built from ordinal less than it and Ω by addition and where , so it’s . What’s more, for .

Then is the 2nd ordinal that cannot be built from ordinal less than it and Ω by addition and where – it’s . What’s more, for β < Ω and .

, and . And for . Note that for any α!

The next step is . Since , it can be built from ordinals below it and Ω and Ω_{2} by addition and where , so it’s not – there’re many things between them, such as , , , , etc.

Then we turn to . For β < Ω_{2} and , but there’re many things between and .

Beyond θ(Ω_{ω},0)

The “getting an uncountable cardinal” in definition start to works now. is the supremum of all the for natural number n.

is not because we have , then , then in . In fact, is the (1+β)-th fixed point of .

Also, is not ; it’s .

The limit of θ function is the supremum of , etc.

For collapsing weakly inaccessible cardinals:

psi_I(a) = a+1-th fixed point of W_W_….

I is the cardinals that used in psi_I() or over.

I_n is the next weakly inaccessible cardinal than I_n-1.

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