In set theory, 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 (Reinhardt 1967, 1974) by American mathematician William Nelson Reinhardt (1939–1998).
A Reinhardt cardinal is the critical point of a non-trivial elementary embedding
This definition refers explicitly to the proper class
In standard ZF, classes are of the form
But it was shown in Suzuki (1999) that no such class is an elementary embedding
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
to the language of ZF, together with axioms stating that
, and Separation and Collection axioms for all formulas involving
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 (1971) proved his inconsistency theorem, showing that the existence of an elementary embedding
contradicts NBG with the axiom of choice (and ZFC extended by
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
Kunen's theorem is not simply a consequence of Suzuki (1999), as it is a consequence of NBG, and hence does not require the assumption that
exists, then there is an elementary embedding of a transitive model
of ZFC (in fact Goedel's constructible universe
There are some variations of Reinhardt cardinals, forming a hierarchy of hypotheses asserting the existence of elementary embeddings
A super Reinhardt cardinal is
[1] The following axioms were introduced by Apter and Sargsyan:[2] J3: There is a nontrivial elementary embedding
J2: There is a nontrivial elementary embedding
is the least fixed-point above the critical point.
as in J1 is known as a super Reinhardt cardinal.
Berkeley cardinals are stronger large cardinals suggested by Woodin.