Abstract

In this talk I will survey the general theory of Boolean ultrapowers, starting from the beginnings and including many applications and some possible future developments. Also, the set-theoretic approach to Boolean ultrapowers, due to recent work of Hamkins and Seabold, will be discussed.

First developed by Mansfield as a purely algebraic construction, Boolean ultrapowers are a natural generalization of usual power-set ultrapowers. More specifically, I will focus on how some combinatorial properties of a ultrafilter U are related to the realization of types in the resulting Boolean ultrapower. Many results on $lambda$-regular and $lambda$-good ultrafilters, mostly due to Keisler, can be generalized to this context. In particular, I will sketch the construction of a $lambda$-good ultrafilter on the Levy collapsing algebra $mathrm{Coll}(lambda,