Farthest-reaching OCF competition

[Moosey]

I swear, we should just have a subforum of the sandbox: "googology stuff"

Anyways, my first submission: a Mahlo-collapsing OCF:

After learning about mahlos on the googology discord:

Code: Select all

B_0,0(a,b) = {0,M} U b
B_m,n+1(a,b) = B_m,n(a,b) U {a+b,ab,a^b,chi_m(h),chi_o(a);a,b,h in B_n,n(a,b),o<m}
B_m(a,b) = U(n<w) B_m,n(a,b)
chi_m(a) = min b|b= intersection(M,B(a,b)), b is regular, b is uncountable

C_0,0(α)={0,Ω,M}
C_n+1,0(α)={γ+δ,γδ,γ^δ,φ_γ(δ),(ψ_0,0(η)),w_γ,chi_γ(δ)|γ,δ,η∈(C_n,0(α));η<α}
C,0(α)=⋃(n<ω)C_n,0(α)
ψ_0,0(α)=min{β∈Ω|β∉C,0(α)}
ψ_1,0(α)=min{β>Ω|β∉C,0(α)}
ψ_2,0(α)=sup(C,0(α))
C_0,m(α)={0,Ω}
C_n+1,m+1(α)={γ+δ,γδ,γ^δ,φ_γ(δ),(ψ_0,m+1(η)),(ψ_2,o(γ)),(ψ_1,o(γ)),(ψ_0,o(γ)),w_γ,chi_γ(δ)| γ,δ,η∈(C_n,m+1(α));η<α,o<(m+1)}
C,m(α)=⋃(n<ω)(C_n,m(α))
ψ_0,m(α)=min{β∈Ω|β∉(C,m(α))}
ψ_1,m(α)=min{β>Ω|β∉(C,m(α))}
ψ_2,m(α)=sup((C,m(α)))

How crappy is it?*

*admittedly it's a big leap from my best OCF previously, but...*
*also, sorry about the sudden change in style between psi and chi--I copied and modified modified a psi I defined on gwiki*
*WHAT ARE YOU DOING READING FINE PRINT?!? PLEASE TELL ME YOU ZOOMED IN TO READ THIS!!!

EDIT: better version from discord:

Code: Select all

B_0,0(a,b) = {0,M} U b
B_m,n+1(a,b) = B_m,n(a,b) U {y+d,yd,y^d,chi_m(h),chi_o(y);y,d,h in B_m,n(a,b),o<m,h<b}
B_m(a,b) = U(n<w) B_m,n(a,b)
chi_m(a) = min b|b= intersection(M,B_m(a,b)), b is regular, b is uncountable

C_0,0(α)={0,Ω,M}
C_n+1,0(α)={γ+δ,γδ,γ^δ,φ_γ(δ),(ψ_0,0(η)),w_γ,chi_γ(δ)|γ,δ,η∈(C_n,0(α));η<α}
C,0(α)=⋃(n<ω)C_n,0(α)
ψ_0,0(α)=min{β∈Ω|β∉C,0(α)}
ψ_1,0(α)=min{β>Ω|β∉C,0(α)}
ψ_2,0(α)=sup(C,0(α))
C_0,m(α)={0,Ω}
C_n+1,m+1(α)={γ+δ,γδ,γ^δ,φ_γ(δ),(ψ_0,m+1(η)),(ψ_2,o(γ)),(ψ_1,o(γ)),(ψ_0,o(γ)),w_γ,chi_γ(δ)| γ,δ,η∈(C_n,m+1(α));η<α,o<(m+1)}
C,m(α)=⋃(n<ω)(C_n,m(α))