Reinhardt cardinal
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).
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 for some set a and formula φ, but in the language of set theory it is not possible to quantify over all classes or define the truth of 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.)
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