# User:Moosey/Ordinals

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Various ordinal related things

Note: for fast-growing functions based on ordinals, See here

## Mahlo-collapsing OCF

(See here)

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))