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Ultrapowers without the axiom of choice

Published online by Cambridge University Press:  12 March 2014

Mitchell Spector
Mitchell Spector*
Affiliation:
Department of Software Engineering and Computer Science, Seattle University, Seattle, Washington 98122
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Abstract

A new method is presented for constructing models of set theory, using a technique of forming pseudo-ultrapowers. In the presence of the axiom of choice, the traditional ultrapower construction has proven to be extremely powerful in set theory and model theory; if the axiom of choice is not assumed, the fundamental theorem of ultrapowers may fail, causing the ultrapower to lose almost all of its utility. The pseudo-ultrapower is designed so that the fundamental theorem holds even if choice fails; this is arranged by means of an application of the omitting types theorem. The general theory of pseudo-ultrapowers is developed. Following that, we study supercompactness in the absence of choice, and we analyze pseudo-ultrapowers of models of the axiom of determinateness and various infinite exponent partition relations. Relationships between pseudo-ultrapowers and forcing are also discussed.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 53 , Issue 4 , December 1988 , pp. 1208 - 1219
Copyright
Copyright © Association for Symbolic Logic 1988

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