Reinhardt cardinal
In set theory, a branch of mathematics, a Reinhardt cardinal is a kind of large cardinal. Reinhardt cardinals are considered under ZF (Zermelo–Fraenkel set theory without the Axiom of Choice), because they are inconsistent with ZFC (ZF with the Axiom of Choice). They were suggested by American mathematician William Nelson Reinhardt (1939–1998). Definition. A Reinhardt cardinal is the critical point of a non-trivial elementary embedding formula_1 of "formula_2" into itself. This definition refers explicitly to the proper class formula_3. In standard ZF, classes are of the form formula_4 for some set formula_5 and formula formula_6. But  it was shown in that no such class is an elementary embedding formula_1. So Reinhardt cardinals are inconsistent with this notion of class. There are other formulations of Reinhardt cardinals which are not known to be inconsistent. One is to add a new function symbol formula_3 to the language of ZF, together with axioms stating that formula_3 is an elementary embedding of formula_2, and Separation and Collection axioms for all formulas involving formula_3. Another is to use a class theory such as NBG or KM, which admit classes which need not be definable in the sense above. Kunen's inconsistency  theorem. proved his inconsistency theorem, showing that the existence of an elementary embedding formula_1 contradicts NBG with the axiom of choice (and ZFC extended by formula_3). 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 "formula_3" and its attendant axioms). Kunen's theorem is not simply a consequence of , as it is a consequence of NBG, and hence does not require the assumption that formula_3 is a definable class. Also, assuming formula_16 exists , then there is an elementary embedding of a transitive model formula_17 of ZFC (in fact Goedel's constructible universe formula_18) into itself. But such embeddings are not classes of formula_17. Stronger axioms. There are some variations of Reinhardt cardinals, forming a hierarchy of hypotheses asserting the existence of elementary embeddings formula_20. A super Reinhardt cardinal is formula_21 such that for every ordinal formula_22, there is an elementary embedding formula_23 with formula_24 and having critical point formula_21. J3: There is a nontrivial elementary embedding formula_26 J2: There is a nontrivial elementary embedding formula_26 and DC holds, where formula_28 is the least  fixed-point above the critical point. J1: Each of J1 and J2 immediately imply J3. A cardinal formula_21 as in J1 is known as a super Reinhardt cardinal. Berkeley cardinals are stronger large cardinals suggested by Woodin.