# Difference between revisions of "User:Moosey/Ordinals"

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(Created page with "Various ordinal related things Note: for fast-growing functions based on ordinals, See here ==Mahlo-collapsing OCF== See [https://www.conwaylife.co...") |
(Added →Mahlo-collapsing OCF) |
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==Mahlo-collapsing OCF== | ==Mahlo-collapsing OCF== | ||

− | See [https://www.conwaylife.com/forums/viewtopic.php?f=12&t=4207 here] | + | (See [https://www.conwaylife.com/forums/viewtopic.php?f=12&t=4207 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)) |

## Revision as of 15:50, 20 November 2019

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))