第20章 莱茵哈特基数（英）

A cardinal κ is Reinhardt if there is an elementary embedding

j : V → V

with critical point κ.

Theorem (Kunen， 1971)

ZFC implies that Reinhardt cardinals don’t exist. In fact， there is

no non-trivial elementary embedding

j : Vλ+2 → Vλ+2

Definition

κ is super Reinhardt if for all ordinals λ there exists a non-trivial

elementary embedding j : V → V such that crit(j)=κ and

j(κ)>λ.

If A is a proper class， then κ is A -super Reinhardt if for all ordinals

λ there exists a non-trivial elementary embedding j : V → V such

that crit(j)=κ， j(κ)>λ， and j(A)= A， where

j(A):= S

α∈OR j(A ∩ Vα).

κ is totally Reinhardt if for each A ∈ Vκ+1，

hVκ， Vκ+1i |= ZF2 +“There is an A -super Reinhardt cardinal”