## FANDOM

10,822 Pages

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, λX > λY, λx(X) > λx(Y) for all x, λX(x) > λY(x) for all x, and λx(Ψ(y,X)) > λx(Ψ(y,Y)) for all x and y.
• If λx(X) is inside of function F(y) = Ψ(y,x), then λx(X) = X.
• Ψ(x,0) = Ψ(x)
• Ψ(x) = Ψε0(x)
• Ψx(0) = x
• Ψx(y+1) = sup(Ψx(y),Ψx(y)Ψx(y)x(y)Ψx(y)Ψx(y),...)
• For limit ordinal y, Ψx(y,z) = sup(Ψx(y[1],z),Ψx(y[2],z),Ψx(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) = ΨΨ(x,1)(y)
• Ψ(0,x+1) = λx
• 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(@λx+1(@2λ0),y),1) (0 < x ≤ y) is the diagonalizer of function F(z) = λx(Ψ(z,y)), in Ψ(Ψ(\$F(z),y),1)
• Ψ(Ψ(@λy(@2λx),x),1) (y ≤ x) is the diagonalizer of function F(z) = λz(Ψ(\$2y,x)), in Ψ(\$F(z),1)
• If X is the diagonalizer of function F(), Ψ(Ψ(@λz(@2X),y),1) (z ≤ y) is the diagonalizer of function G(w) = Ψ(\$2λz(F(w)),y), in Ψ(Ψ(\$λz(\$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(Ψλ0(0),1)).
• The limit of sDAN is Ψ(Ψ(Ψ(1,1)λ0ω,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)Ψλ0(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(ΨΨ(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.