M到K有点复杂,就不说了。在说I和M之前,先简单说一下Ω
ψ(...)是递归序数,Ω是ψ(...)的上确界,ψ(α)<Ω对任意α成立
ψ_1(...)是关于Ω的递归序数,它的上确界是Ω_2,对任意α,ψ_1(α)<Ω_2
同理,ψ_2(...)是基于Ω_2的递归序数,它的上确界是Ω_3
Ω_ψ(...)的上确界是Ω_Ω
Ω_ψ_1(...)的上确界是Ω_Ω_2
ψ_I(...)是α趋于Ω_α的不动点(下文简称Ω_α不动点),I是ψ_I(...)的上确界,对任意α,ψ_I(α)<I
ψ_{I+1}(...)是基于I的递归序数,它的上确界是Ω_(I+1)
ψ_I_2(...)是I之后的Ω_α不动点,I_2是ψ_I_2(...)的上确界
ψ_I_3(...)是I_2之后的Ω_α不动点,I_3是ψ_I_3(...)的上确界
然后有I_4、I_ω、I_Ω、I_I、I_I_I等,直到ψ_I(1,0)(0)
ψ_I(1,0)(...)是I_α不动点,I(1,0)是ψ_I(1,0)(...)的上确界
ψ_{I(1,0)+1}(...)是基于I(1,0)的递归序数,它的上确界是Ω_(I(1,0)+1)
ψ_{I_(I(1,0)+1)}(...)是I(1,0)之后的Ω_α不动点,它的上确界是I_(I(1,0)+1)
ψ_I(1,1)(...)是I(1,0)之后的I_α不动点,I(1,1)是ψ_I(1,1)(...)的上确界
ψ_I(1,2)(...)是I(1,1)之后的I_α不动点,I(1,2)是ψ_I(1,2)(...)的上确界
然后有I(1,3)、I(1,ω)、I(1,Ω)、I(1,I)、I(1,I(1,0))、I(1,I(1,I(1,0)))等,直到ψ_I(2,0)(0)
ψ_I(2,0)(...)是I(1,α)不动点,I(2,0)是ψ_I(2,0)(...)的上确界
ψ_I(2,1)(...)是I(2,0)之后的I(1,α)不动点,它的上确界是I(2,1)
ψ_I(3,0)(...)是I(2,α)不动点,I(3,0)是ψ_I(3,0)(...)的上确界
然后有I(ω,0)、I(Ω,0)、I(I,0)、I(I(1,0),0)、I(I(2,0),0)、I(I(I,0),0)、I(I(I(I,0),0),0)...直到ψ_I(1,0,0)(0)
直到ψ_I(1,0,0)(...)是I(α,0)不动点,它的上确界是I(1,0,0)
ψ_{I(1,0,0)+1}(...)是基于I(1,0,0)的递归序数,它的上确界是Ω_(I(1,0,0)+1)
ψ_{I_(I(1,0,0)+1)}(...)是I(1,0,0)之后的Ω_α不动点,它的上确界是I_(I(1,0,0)+1)
ψ_{I(1,I(1,0,0)+1)}(...)是I(1,0,0)之后的I_α不动点,它的上确界是I(1,I(1,0,0)+1)
然后有I(1,0,0)之后的I(2,α)不动点、I(ω,α)不动点等
ψ_I(1,0,1)(...)是I(1,0,0)之后的I(α,0)不动点,它的上确界是I(1,0,1)
ψ_I(1,1,0)(...)是I(1,0,α)不动点,它的上确界是I(1,1,0)
ψ_I(2,0,0)(...)是I(1,α,0)不动点,它的上确界是I(2,0,0)
ψ_I(1,0,0,0)(...)是I(α,0,0)不动点,它的上确界是I(1,0,0,0)
然后有I(1,0,0,0,0)、I(1,0,0,0,0,0)等,而M在所有这一切之上
I(1,0)、I(1,0,0)、I(1@ω)之于M,大致相当于ε_0、Γ_0、SVO之于Ω
ψ(...)是递归序数,Ω是ψ(...)的上确界,ψ(α)<Ω对任意α成立
ψ_1(...)是关于Ω的递归序数,它的上确界是Ω_2,对任意α,ψ_1(α)<Ω_2
同理,ψ_2(...)是基于Ω_2的递归序数,它的上确界是Ω_3
Ω_ψ(...)的上确界是Ω_Ω
Ω_ψ_1(...)的上确界是Ω_Ω_2
ψ_I(...)是α趋于Ω_α的不动点(下文简称Ω_α不动点),I是ψ_I(...)的上确界,对任意α,ψ_I(α)<I
ψ_{I+1}(...)是基于I的递归序数,它的上确界是Ω_(I+1)
ψ_I_2(...)是I之后的Ω_α不动点,I_2是ψ_I_2(...)的上确界
ψ_I_3(...)是I_2之后的Ω_α不动点,I_3是ψ_I_3(...)的上确界
然后有I_4、I_ω、I_Ω、I_I、I_I_I等,直到ψ_I(1,0)(0)
ψ_I(1,0)(...)是I_α不动点,I(1,0)是ψ_I(1,0)(...)的上确界
ψ_{I(1,0)+1}(...)是基于I(1,0)的递归序数,它的上确界是Ω_(I(1,0)+1)
ψ_{I_(I(1,0)+1)}(...)是I(1,0)之后的Ω_α不动点,它的上确界是I_(I(1,0)+1)
ψ_I(1,1)(...)是I(1,0)之后的I_α不动点,I(1,1)是ψ_I(1,1)(...)的上确界
ψ_I(1,2)(...)是I(1,1)之后的I_α不动点,I(1,2)是ψ_I(1,2)(...)的上确界
然后有I(1,3)、I(1,ω)、I(1,Ω)、I(1,I)、I(1,I(1,0))、I(1,I(1,I(1,0)))等,直到ψ_I(2,0)(0)
ψ_I(2,0)(...)是I(1,α)不动点,I(2,0)是ψ_I(2,0)(...)的上确界
ψ_I(2,1)(...)是I(2,0)之后的I(1,α)不动点,它的上确界是I(2,1)
ψ_I(3,0)(...)是I(2,α)不动点,I(3,0)是ψ_I(3,0)(...)的上确界
然后有I(ω,0)、I(Ω,0)、I(I,0)、I(I(1,0),0)、I(I(2,0),0)、I(I(I,0),0)、I(I(I(I,0),0),0)...直到ψ_I(1,0,0)(0)
直到ψ_I(1,0,0)(...)是I(α,0)不动点,它的上确界是I(1,0,0)
ψ_{I(1,0,0)+1}(...)是基于I(1,0,0)的递归序数,它的上确界是Ω_(I(1,0,0)+1)
ψ_{I_(I(1,0,0)+1)}(...)是I(1,0,0)之后的Ω_α不动点,它的上确界是I_(I(1,0,0)+1)
ψ_{I(1,I(1,0,0)+1)}(...)是I(1,0,0)之后的I_α不动点,它的上确界是I(1,I(1,0,0)+1)
然后有I(1,0,0)之后的I(2,α)不动点、I(ω,α)不动点等
ψ_I(1,0,1)(...)是I(1,0,0)之后的I(α,0)不动点,它的上确界是I(1,0,1)
ψ_I(1,1,0)(...)是I(1,0,α)不动点,它的上确界是I(1,1,0)
ψ_I(2,0,0)(...)是I(1,α,0)不动点,它的上确界是I(2,0,0)
ψ_I(1,0,0,0)(...)是I(α,0,0)不动点,它的上确界是I(1,0,0,0)
然后有I(1,0,0,0,0)、I(1,0,0,0,0,0)等,而M在所有这一切之上
I(1,0)、I(1,0,0)、I(1@ω)之于M,大致相当于ε_0、Γ_0、SVO之于Ω