**April 14**

**Gabriel Goldberg**,
University of California, Berkeley

**Cardinal preserving embeddings and strongly compact cardinals**

Kunen's theorem that there is no elementary embedding from V to V seems to set an upper bound on the hierarchy of large cardinal axioms. Challenging this, Caicedo asked what happens when V is replaced with an inner model M that is very close to V in the sense that M correctly computes the class of cardinals. Assuming the existence of strongly compact cardinals, we show that there is no elementary embedding from such an inner model M into V or from V into M. The former result (M into V) is joint work with Sebastiano Thei. Without strong compactness assumptions, both questions remain open, but we'll discuss some partial results.