**October 16**

Richard Matthews,
University of Leeds

**Taking Reinhardt's Power Away**

Many large cardinals can be defined through elementary embeddings from the set-theoretic universe to some inner model, with the guiding principle being that the closer the inner model is to the universe the stronger the resulting theory. Under ZFC, the Kunen Inconsistency places a hard limit on how close this can be. One is then naturally led to the question of what theory is necessary to derive this inconsistency with the primary focus having historically been embeddings in ZF without Choice.

In this talk we take a different approach to weakening the required theory, which is to study elementary embeddings from the universe into itself in ZFC without Power Set. We shall see that I1, one of the largest large cardinal axioms not known to be inconsistent with ZFC, gives an upper bound to the naive version of this question. However, under reasonable assumptions, we can reobtain this inconsistency in our weaker theory.