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Some size and structure theorems for ultrapowers

University of British Columbia

In this thesis we study the mapping D → A[sup I]/D, between ultrafilters and models, given by the ultrapower construction. Under this mapping homomorphisms of ultrapowers induce elementary embeddings of ultrapowers. Using these embeddings we investigate the dependence of the structure of an ultrapower A[sup I]/D on the cardinality of the index set I. With each ultrafilter D we associate a set of cardinals a(D) which we term the shadow of D. We investigate the form of the sets a(D). It is shown that if σ(D) has "gaps" then isomorphisms arise between ultrapowers of different index sizes. In terms of σ(D) we prove new results on the properties of the set of homomorphic images of an ultrafilter. Finally we introduce a new class of "quasicomplete" ultrafilters and prove several results about ultrapowers constructed using these. Two results which can be mentioned here are the following: Let α be a regular cardinal. We establish necessary and sufficient conditions on D (i) for the cardinality of α to be raised in the passage to α[sup I]/D. (ii) for the confinality of α[sup I]/D (regarded as an ordered set) to be greater than α⁺. Some of the results of this thesis depend on assumption of the Generalised Continuum Hypothesis. The result (i) above is a case in point.

Text

2011-03-21

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http://hdl.handle.net/2429/32660

Mathematics

University of British Columbia

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