Journal of Symbolic Logic
- J. Symbolic Logic
- Volume 53, Issue 4 (1988), 1208-1219.
Ultrapowers Without the Axiom of Choice
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.
J. Symbolic Logic, Volume 53, Issue 4 (1988), 1208-1219.
First available in Project Euclid: 6 July 2007
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Spector, Mitchell. Ultrapowers Without the Axiom of Choice. J. Symbolic Logic 53 (1988), no. 4, 1208--1219. https://projecteuclid.org/euclid.jsl/1183742791