In mathematical set theory, a Reinhardt cardinal is a large cardinal κ in a model of ZF, Zermelo–Fraenkel set theory without the axiom of choice (Reinhardt cardinals are not compatible with the axiom of choice in ZFC). They were suggested by William Nelson Reinhardt (1967, 1974).
Contents
Definition
A Reinhardt cardinal is the critical point of a non-trivial elementary embedding j of V into itself.
A minor technical problem is that this property cannot be formulated in the usual set theory ZFC: the embedding j is a class, which in ZFC means something of the form
Kunen's theorem
Kunen (1971) proved Kunen's inconsistency theorem showing that the existence of such an embedding contradicts NBG with the axiom of choice (and ZFC extended by j), but it is consistent with weaker class theories. His proof uses the axiom of choice, and it is still an open question as to whether such an embedding is consistent with NBG without the axiom of choice (or with ZF plus the extra symbol j and its attendant axioms).
Stronger cardinal axioms
There are some variations of Reinhardt cardinals. In ZF, there is a hierarchy of hypotheses asserting existence of elementary embeddings V→V
J3: There is a nontrivial elementary embedding j: V→V
J2: There is a nontrivial elementary embedding j: V→V, and DCλ holds, where λ is the least fixed-point above the critical point.
J1: There is a cardinal κ such that for every α, there is an elementary embedding j : V→V with j(κ)>α and cp(j) = κ.
J2 implies J3, and J1 implies J3 and also implies consistency of J2. By adding a generic well-ordering of V to a model of J1, one gets ZFC plus a nontrivial elementary embedding of HOD into itself.
Berkeley cardinals are a stronger large cardinals suggested by Woodin.