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))


Extension with a-Mahlos:

B_p,n,m(x,y) = {0,J}Uy, where J is the smallest p-Mahlo
B_p,n,m+1(x,y) = {a+b,theta_p,0(h),theta_q,o(a)|a,b,h in C_p,n,m(a,b),q<p,o<n,h<y}
B_p,n(x,y) = U(m<w) C_p,n,m(x,y)
Theta_p+1,n(x) = min b|b=intersection(J,B_p,n(x,b)),b is p-Mahlo, J is the smallest p+1-mahlo
Theta_0,n(x) = min b|b=intersection(M,B_p,n(x,b)),b is Regular and uncountable

C_0,m(α)={0,Ω,M}
C_n+1,0(α)={γ+δ,γδ,γ^δ,φ_γ(δ),(ψ_0,0(η)),w_γ,theta_{γ,0}(δ),minimum γ-Mahlo|γ,δ,η∈(C_n,0(α));η<α}
C,0(α)=⋃(n<ω)C_n,0(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))
C_n+1,m(α)={γ+δ,γδ,γ^δ,φ_γ(δ),(ψ_0,m(η)),(ψ_2,o(γ)),(ψ_1,o(γ)),(ψ_0,o(γ)),w_γ,theta_{γ,o}(δ),minimum γ-Mahlo|γ,δ,η∈(C_n,m(α));η<α,o<(m)}
C,m(α)=⋃(n<ω)(C_n,m(α))