Asaf Karagila (University of Leeds)

How large must a small measurable cardinal must be?

Every measurable cardinal is inaccessible. At least assuming the Axiom of Choice. Without the Axiom of Choice, however, we can have measurable cardinals which are successor cardinals. Discounting those, how "small" on the large cardinal hierarchy can the least measurable cardinal get? It turns out that the answer depends on available large cardinals in inner models.

Specifically, in a recent work with Gitik and Hayut, we show that if the least measurable is the least inaccessible cardinal, then in the core model there is a measurable of a relatively high Mitchell order. This is in contrast to an older work with Hayut where we show that for the least measurable to be the least Mahlo cardinal we only need a single measurable cardinal in an inner model.