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).

DefinitionEdit

A Reinhardt cardinal is the critical point of a non-trivial elementary embedding   of   into itself.

This definition refers explicitly to the proper class  . In standard ZF, classes are of the form   for some set   and formula  . 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   is an elementary embedding of  , 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's inconsistency theoremEdit

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   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   is a definable class. Also, assuming   exists, then there is an elementary embedding of a transitive model   of ZFC (in fact Goedel's constructible universe  ) into itself. But such embeddings are not classes of  .

Stronger axiomsEdit

There are some variations of Reinhardt cardinals, forming a hierarchy of hypotheses asserting the existence of elementary embeddings  .

A super Reinhardt cardinal is   such that for every ordinal  , there is an elementary embedding   with   and having critical point  .

J3: There is a nontrivial elementary embedding  
J2: There is a nontrivial elementary embedding   and DC  holds, where   is the least fixed-point above the critical point.
J1:

Each of J1 and J2 immediately imply J3. A cardinal   as in J1 is known as a super Reinhardt cardinal.

Berkeley cardinals are stronger large cardinals suggested by Woodin.

See alsoEdit

ReferencesEdit

  • Jensen, Ronald (1995), "Inner Models and Large Cardinals", The Bulletin of Symbolic Logic, The Bulletin of Symbolic Logic, Vol. 1, No. 4, 1 (4): 393–407, CiteSeerX 10.1.1.28.1790, doi:10.2307/421129, JSTOR 421129
  • Kanamori, Akihiro (2003), The Higher Infinite : Large Cardinals in Set Theory from Their Beginnings (2nd ed.), Springer, ISBN 3-540-00384-3
  • Kunen, Kenneth (1971), "Elementary embeddings and infinitary combinatorics", Journal of Symbolic Logic, The Journal of Symbolic Logic, Vol. 36, No. 3, 36 (3): 407–413, doi:10.2307/2269948, JSTOR 2269948, MR 0311478
  • Reinhardt, W. N. (1967), Topics in the metamathematics of set theory, Doctoral dissertation, University of California, Berkeley
  • Reinhardt, W. N. (1974), "Remarks on reflection principles, large cardinals, and elementary embeddings.", Axiomatic set theory, Proc. Sympos. Pure Math., vol. XIII, Part II, Providence, R. I.: Amer. Math. Soc., pp. 189–205, MR 0401475
  • Suzuki, Akira (1999), "No elementary embedding from V into V is definable from parameters", Journal of Symbolic Logic, 64 (4): 1591–1594, doi:10.2307/2586799, JSTOR 2586799, MR 1780073

External linksEdit