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Reinhardt cardinal

Summary

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 (Reinhardt 1967, 1974) by American mathematician William Nelson Reinhardt (1939–1998).

Definition

A Reinhardt cardinal is the critical point of a non-trivial elementary embedding ${\displaystyle j:V\to V}$  of ${\displaystyle V}$  into itself.

This definition refers explicitly to the proper class ${\displaystyle j}$ . In standard ZF, classes are of the form ${\displaystyle \{x|\phi (x,a)\}}$  for some set ${\displaystyle a}$  and formula ${\displaystyle \phi }$ . But it was shown in Suzuki (1999) that no such class is an elementary embedding ${\displaystyle j:V\to V}$ . 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 ${\displaystyle j}$  to the language of ZF, together with axioms stating that ${\displaystyle j}$  is an elementary embedding of ${\displaystyle V}$ , and Separation and Collection axioms for all formulas involving ${\displaystyle j}$ . 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

Kunen (1971) proved his inconsistency theorem, showing that the existence of an elementary embedding ${\displaystyle j:V\to V}$  contradicts NBG with the axiom of choice (and ZFC extended by ${\displaystyle j}$ ). 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 ${\displaystyle j}$  and its attendant axioms).

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 ${\displaystyle j}$  is a definable class. Also, assuming ${\displaystyle 0^{\#}}$  exists, then there is an elementary embedding of a transitive model ${\displaystyle M}$  of ZFC (in fact Goedel's constructible universe ${\displaystyle L}$ ) into itself. But such embeddings are not classes of ${\displaystyle M}$ .

Stronger axioms

There are some variations of Reinhardt cardinals, forming a hierarchy of hypotheses asserting the existence of elementary embeddings ${\displaystyle V\to V}$ .

A super Reinhardt cardinal is ${\displaystyle \kappa }$  such that for every ordinal ${\displaystyle \alpha }$ , there is an elementary embedding ${\displaystyle j:V\to V}$  with ${\displaystyle j(\kappa )>\alpha }$  and having critical point ${\displaystyle \kappa }$ .

J3: There is a nontrivial elementary embedding ${\displaystyle j:V\to V}$
J2: There is a nontrivial elementary embedding ${\displaystyle j:V\to V}$  and DC${\displaystyle \lambda }$  holds, where ${\displaystyle \lambda }$  is the least fixed-point above the critical point.
J1:

Each of J1 and J2 immediately imply J3. A cardinal ${\displaystyle \kappa }$  as in J1 is known as a super Reinhardt cardinal.

Berkeley cardinals are stronger large cardinals suggested by Woodin.