In set theory, a branch of mathematics, a Reinhardt cardinal is a large cardinal κ, suggested by William Nelson Reinhardt (1967, 1974), that 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 of the form for some set a and formula φ, and in the language of set theory it is not possible to quantify over all classes (or formulas). There are several ways to get round this. One way is to add a new function symbol j to the language of ZFC, together with axioms stating that j is an elementary embedding of V (and of course adding separation and replacement axioms for formulas involving j). Another way is to use a class theory such as NBG or KM. A third way would be to treat Kunen's theorem as a countable infinite collection of theorems, one for each formula φ, but that would trivialize the theorem. (It is possible to have nontrivial elementary embeddings of transitive models of ZFC into themselves assuming a mild large cardinal hypothesis, but these elementary embeddings are not classes of the model.)
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).
Reinhardt cardinals are essentially the largest ones that have been defined (as of 2006) that are not known to be inconsistent in ZF set theory.
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.
Woodin also introduced the following large cardinal hypothesis for ZF, which he called Berkeley cardinal:
There is an ordinal κ, such that for every transitive set M that includes κ, there is a nontrivial elementary embedding of M into M with critical point below κ.
It is not known whether this implies consistency of J1. A weakening of being a Berkeley cardinal is that for every binary relation R on Vκ, there is a nontrivial elementary embedding of (Vκ, R) into itself. This implies that that we have elementary j1, j2, j3, ...
j1: (Vκ, ∈) →(Vκ, ∈),
j2: (Vκ, ∈, j1) → (Vκ, ∈, j1),
j3: (Vκ, ∈, j1, j2) → (Vκ, ∈, j1, j2),
and so on. This can be continued any finite number of times, and to the extent that the model has dependent choice, transfinitely. Thus, plausibly, this notion can be strengthened simply by asserting more dependent choice.
While all these notions are incompatible with ZFC, their consequences do not appear to be false. There is no known inconsistency with ZFC in asserting that, for example:
For every ordinal λ, there is a transitive model of ZF + Berkeley cardinal that is closed under λ sequences.
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