Nontrivial elementary embedding with a definable bounding of the universe into itself
by
Akira Suzuki
Kobe University

Kunen showed the non-existence of nontrivial elementary embeddings of V into itself with the AC. We show, without the AC, that there is no nontrivial elementary embedding with a definable bounding of V into itself. We say b: Ord ® Ord is a bounding for j:V ® V if rank(j(a)) £ b(a) for each ordinal a. In this talk, we say a mapping b: Ord ® Ord is definable when b is definable in (V; Î ) with arbitrary parameter p Î V.

Date received: February 25, 2005