Jonas Eliasson
CMU and Uppsala, Sweden

"Ultrapowers as sheaves"

ABSTRACT: Given a set S and an ultrafilter (I,U) we can form the ultrapower S^I/U of S over (I,U). But what happens when you vary the ultrafilter U? An answer to this question has been given by Andreas Blass in the study of ultrafilters under the Rudin-Keisler ordering. In this talk I will show how these varied ultrapowers can be seen as sheaves on a category of ultrafilters.

I will introduce the collection of sheaves on ultrafilters (a topos) and show how it relates to the ultrapowers. I will explain how this arose from the constructive sheaf model of nonstandard arithmetic introduced by Ieke Moerdijk, and show how the topos can be used to model nonstandard set theory.

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