Metastages
Prologue
Uncountables
Before I start define Metastages, let take a look for my OCF first called TehAarex's OCF (TAOCF). Let Sx is equal to the successor ordinal of x.
- 0 = a smallest ordinal
- ψ(0) = a smallest ordinal that isn't constructible by adding 0s.
- ψ(Sx) = a smallest ordinal that isn't constructible by adding ψ(x)s.
The limit of TAOCF without cardinals is the fixed point of a → ψ(a).
Define the first cardinal Ω as a diagonalizer of ψ(_). But what is a diagonalizer for Ω?
Let cof be a function that:
- cof(Sx) = 1
- cof(psi(Sx)) = w
- cof(psi(x)) = cof(x)
- cof(Ω) = Ω
If cof(x) = Ω, then psi(x) = a smallest ordinal that isn't constructible by using all ψ() terms that is less than ψ(x).
Then I define ψ_Ω_x() function and Ω_x cardinals. Let ψ_Ω() be the same function as ψ() and ψ_Ω_Sx() act like ψ() but however ψ_Ω_Sx(0) is actually a smallest ordinal that isn't constructible by adding Ω_x. Define Ω_n as a diagonalizer of ψ_Ω_n(_) such that:
- If cof(x) = Ω_n, then ψ_Ω_n(x) = a smallest ordinal that isn't constructible by using all ψ_Ω_n() terms that is less than ψ_Ω_n(x).
Define 2 new cof rules:
- cof(Ω_Sn) = Ω_Sn
- cof(Ω_n) = cof(n)
Weakly Inaccessibles
I define new cardinals that goes beyond uncountable cardinals. First, let all terms of Ω_x be the set of uncountable cardinals called UΩ.
Let Ω_Sx be a next uncountable cardinal that comes after the fixed point of a → Ω_x+a. So psi_Ω_Sx enumerate the fixed points of a → Ω_x+a. Furthermore, I is a first uncountable cardinal that comes after the fixed point of a → W_a then I can define psi_I such that it enumerate the fixed points of a → Ω_a.
[MORE COMING SOON]