Ultrapowers without the axiom of choice

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

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Citations of this work

On a Spector Ultrapower for the Solovay Model.Vladimir Kanovei & Michiel van Lambalgen - 1997 - Mathematical Logic Quarterly 43 (3):389-395.
Iterated extended ultrapowers and supercompactness without choice.Mitchell Spector - 1991 - Annals of Pure and Applied Logic 54 (2):179-194.

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References found in this work

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 Mathematical 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.

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