Making an OCF, attempt 2

I got inspired by this blog post to make another OCF, or at least something similar to it. I will call my previous OCF the JKL OCF and this one the lambda notation.

• @_x is the rest of the ordinal inside of the function the @_x is inside of, excluding ends of parentheses...
• ...and $_x is a copy of that @_x (if $_x is empty, the last $_x in an expression is replaced to 0)
• @₁ = @ and $₁ = $
• A function F() eventually overgrows another function G() if there's some value y < α -> F(α) such that F(x) ≥ G(x) for all x ≥ y
• If X is a diagonalizer of F(), @X = sup($,$F($),$F($F($)),...)
• If X is a diagonalizer of F(), Y is a diagonalizer of G(), and F() eventually overgrows G(), then X > Y
• If X > Y, then @X > $Y, λ_X > λ_Y, λ_x(X) > λ_x(Y) for all x, λ_X(x) > λ_Y(x) for all x, and λ_x(Ψ(y,X)) > λ_x(Ψ(y,Y)) for all x and y.
• If λ_x(X) is inside of function F(y) = Ψ(y,x), then λ_x(X) = X.
• Ψ(x,0) = Ψ(x)
• Ψ(x) = Ψ_ε0(x)
• Ψ_x(0) = x
• Ψ_x(y+1) = sup(Ψ_x(y),Ψ_x(y)^Ψ_x(y),Ψ_x(y)^{Ψ_x(y)Ψ_x(y)},...)
• For limit ordinal y, Ψ_x(y,z) = sup(Ψ_x(y[1],z),Ψ_x(y[2],z),Ψ_x(y[3],z),...)
• To avoid ellipses in the following rules, Ψ(Ψ(a,b),c) (b > c) is shortened to Ψ(a,c) (and Ψ(Ψ(a,b),1) means Ψ(a,1) if b = 1), but only in the rules.
• λ₀ is the diagonalizer of the function F(x) = Ψ(x,0)
• Ψ(x+1,1) is the diagonalizer of the function F(y) = Ψ_Ψ(x,1)(y)
• Ψ(0,x+1) = λ_x
• Ψ(Ψ(@λ₀(x+λ₀),y),1) is the diagonalizer of function F(z) = Ψ(Ψ(Ψ($λ₀(x)+z,y),1))
• Ψ(Ψ(@λ₀(@₂λ₀),x),1) is the diagonalizer of function F(y) = Ψ(Ψ(Ψ($λ₀($₂y),x),1))
• Ψ(Ψ(@λ_x+1(@₂λ₀),y),1) (0 < x ≤ y) is the diagonalizer of function F(z) = λ_x(Ψ(z,y)), in Ψ(Ψ($F(z),y),1)
• Ψ(Ψ(@λ_y(@₂λ_x),x),1) (y ≤ x) is the diagonalizer of function F(z) = λ_z(Ψ($₂y,x)), in Ψ($F(z),1)
• If X is the diagonalizer of function F(), Ψ(Ψ(@λ_z(@₂X),y),1) (z ≤ y) is the diagonalizer of function G(w) = Ψ($₂λ_z(F(w)),y), in Ψ(Ψ($λ_z($₂G(w)),y),1)
• If none of the rules apply, Ψ(x,y) can be changed to F(x) while calculating the FS of the ordinal (but only while doing that), where F(z) = Ψ(z,y)

The current limit of this function is Ψ(Ψ(Ψ(Ψ(...,3),2),1)). I could extend to up to Ψ(Ψ(α -> Ψ(0,α),1)), but this notation is already strong enough. If there is a problem in the definition, please leave it in the comments.

This function has a WAY faster growth rate than the strength I intended the JKL OCF to have. It's also a good notation to compare SAN with, since both notations use the same concept: in SAN it's 1-separators, and in here it's diagonalizers. I have done some analysis with this notation and SAN, and the results are:

• The limit of pDAN is Ψ(Ψ(Ψ_λ0(0),1)).
• The limit of sDAN is Ψ(Ψ(Ψ(1,1)λ₀^ω,1)).
• The limit of DAN is Ψ(Ψ(Ψ(1,1)^ω,1)).
• The limit of NDAN is Ψ(Ψ(Ψ(1,1)^λ₀ω,1)).
• The limit of WDEN is Ψ(Ψ(Ψ(1,1)^Ψ_λ0(0),1)).
• The limit of mWDEN is >> Ψ(Ψ(Ψ(1,1)^Ψ(1,1)ω,1)) (I am not sure how mWDEN works starting from (1^:)(1^:)(1^:)...{1:`2}, but if my guesses are correct, the limit of mWDEN is Ψ(Ψ(Ψ_Ψ(1,1)(0),1)).

You can find the analysis of this notation and standard ordinal notation here. I will make the blog post of the analysis of this notation and SAN after I finish the analysis of this notation and standard ordinal notation, but I can also make it now, if people want.