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Saturated Boolean Ultrapowers

Friday 9th October 2015 - 14:40 to 15:35
INI Seminar Room 2
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 $\mathrm{Coll}(w, <\lambda)$. In addition to that, I will describe a possible application to Keisler's order on complete theories.

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University of Cambridge Research Councils UK
    Clay Mathematics Institute London Mathematical Society NM Rothschild and Sons