Elementary embeddings j: V -> V with Choice
by
Paul Corazza
Boise State University

In his Ph.D. thesis, W. Reinhardt gave plausibility arguments for the ``ultimate'' large cardinal axiom: the existence of a nontrivial elementary embedding j: V --> V. In 1970, K. Kunen showed that the existence of such an embedding leads to an inconsistency, at least if his argument is formalized in Kelley-Morse (KM) set theory. We show here that the conclusion one can actually draw from Kunen's argument depends on the context in which the argument is formalized. When formalized axiomatically in the language {\epsilon, j}, we obtain a finer analysis of these embeddings and show:

1. There is a spectrum of axioms of the form `` existsj: V --> V'' ranging in consistency strength from Con(ZFC) through I₃, with inconsistency arising only as a special case.
2. Kunen's proof requires assumptions that are independent of the theory ZFC+ ``j is a nontrivial elementary embedding'' in order to arrive at inconsistency.

We also show how it is possible to have an elementary embedding L --> L when 0^# fails. We look at natural connections between Schindler's remarkable cardinals and certain types of embeddings j: M --> M. Finally, we bridge the gap between the strongest of the ``j: V --> V'' axioms not known to be inconsistent and the axiom I₃ by defining a sequence of axioms Iⁿ₄(\kappa).

Corazza's Home Page

Date received: January 31, 2001

Copyright © 2001 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cagb-02.