Journal of Symbolic Logic

Ultrapowers Without the Axiom of Choice

Mitchell Spector

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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.

Article information

Source
J. Symbolic Logic, Volume 53, Issue 4 (1988), 1208-1219.

Dates
First available in Project Euclid: 6 July 2007

Permanent link to this document
https://projecteuclid.org/euclid.jsl/1183742791

Mathematical Reviews number (MathSciNet)
MR973110

Zentralblatt MATH identifier
0673.03037

JSTOR
links.jstor.org

Citation

Spector, Mitchell. Ultrapowers Without the Axiom of Choice. J. Symbolic Logic 53 (1988), no. 4, 1208--1219. https://projecteuclid.org/euclid.jsl/1183742791


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