David Bourget (Western Ontario)
David Chalmers (ANU, NYU)
Rafael De Clercq
Ezio Di Nucci
Jack Alan Reynolds
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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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References found in this work BETA
Leo A. Harrington & Alexander S. Kechris (1981). On the Determinacy of Games on Ordinals. Annals of Mathematical Logic 20 (2):109-154.
J. M. Henle (1979). Researches Into the World of K → K. Annals of Mathematical Logic 17 (1-2):151-169.
J. M. Henle (1983). Magidor-Like and Radin-Like Forcing. Annals of Pure and Applied Logic 25 (1):59-72.
Citations of this work BETA
Mitchell Spector (1991). Iterated Extended Ultrapowers and Supercompactness Without Choice. Annals of Pure and Applied Logic 54 (2):179-194.
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