Nontrivial elementary embedding from V into V
by
Akira Suzuki
Department of Computer Science, Ritsumeikan University

In 1970, Kenneth Kunen showed that there is no non-trivial elementary embedding of the universe V into itself using the axiom of choice. Kunen remarked in his paper that the result can be formalized in Morse-Kelley theory of sets and classes. In this talk, we will work within ZF, Zermelo-Fraenkel axioms, and deal with embeddings definable with a formula and a parameter.

In ZF, a ``class'' is usually synonymous with ``property'', that is a class definable with a parameter, C={x  :  j(x, p)}, where j is a formula in the language { in }. Using this convention, let j be a class. Then ``j is an elementary embedding of V into V'' is not a single statement but a schema of statements ``j preserves \psi, '' for each formula \psi. We prove that this schema is expressible in the language { in } by a single formula:

Lemma[ZF]. An embedding j\colon V --> V is elementary iff j preserves \Psi.

Here \Psi(\alpha, j, a) is the property ``\alpha is an ordinal, j is a formula and V_\alpha\modelsj[a].'' The lemma is of course a schema of lemmas, one for each formula defining j and for each \psi to be preserved.

Using this we prove our theorem in ZF (again, a schema of theorems.):

Theorem[ZF]. There is no nontrivial definable elementary embedding j\colon V --> V.

Date received: June 18, 1999

Copyright © 1999 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 # caby-13.