Ultrapowers without the axiom of choice

Journal of Symbolic Logic 53 (4):1208-1219 (1988)

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
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DOI 10.2307/2274614
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References found in this work BETA

Model Theory.Michael Makkai, C. C. Chang & H. J. Keisler - 1991 - Journal of Symbolic Logic 56 (3):1096.
On the Determinacy of Games on Ordinals.L. A. Harrington - 1981 - Annals of Pure and Applied Logic 20 (2):109.
Magidor-Like and Radin-Like Forcing.J. M. Henle - 1983 - Annals of Pure and Applied Logic 25 (1):59-72.
Researches Into the World of "X" [Implies] "X".J. M. Henle - 1979 - Annals of Mathematical Logic 17 (1/2):151.
Model Theory for Infinitary Logic. Logic with Countable Conjunctions and Finite Quantifiers.[author unknown] - 1973 - Journal of Symbolic Logic 38 (3):522-523.

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Iterated Extended Ultrapowers and Supercompactness Without Choice.Mitchell Spector - 1991 - Annals of Pure and Applied Logic 54 (2):179-194.

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