Reinhardt cardinal

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In set theory, a branch of mathematics, a Reinhardt cardinal is a large cardinal κ, suggested by Reinhardt (1967, 1974), that 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: a non-trivial elementary embedding of the universe into itself cannot be coded as a set. 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.

Kunen (1971) showed that the existence of such an embedding contradicts NBG with the axiom of choice (and ZFC extended by j), but it is consistent with weaker class theories. 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 j).

Reinhardt cardinals are essentially the largest ones that have been defined (as of 2006) that are not known to be inconsistent in ZF-set theory.

[edit] References

• Jensen, Ronald (1995), “Inner Models and Large Cardinals”, The Bulletin of Symbolic Logic 1 (4): 393-407., <http://links.jstor.org/sici?sici=1079-8986%28199512%291%3A4%3C393%3AIMALC%3E2.0.CO%3B2-V> 
• Kanamori, Akihiro (2003), The Higher Infinite : Large Cardinals in Set Theory from Their Beginnings (2nd ed ed.), Springer, ISBN 3-540-00384-3 
• Kunen, Kenneth (1971), “Elementary embeddings and infinitary combinatorics.”, J. Symbolic Logic 36: 407-413, MR0311478, <http://links.jstor.org/sici?sici=0022-4812%28197109%2936%3A3%3C407%3AEEAIC%3E2.0.CO%3B2-W> 
• 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, vol. XIII, Part II, Proc. Sympos. Pure Math., Providence, R. I.: Amer. Math. Soc., pp. 189--205, MR0401475