Model theory under the axiom of determinateness

Journal of Symbolic Logic 50 (3):773-780 (1985)

Abstract
We initiate the study of model theory in the absence of the Axiom of Choice, using the Axiom of Determinateness as a powerful substitute. We first show that, in this context, L ω 1 ω is no more powerful than first-order logic. The emphasis then turns to upward Lowenhein-Skolem theorems; ℵ 1 is the Hanf number of first-order logic, of L ω 1 ω , and of a strong fragment of L ω 1 ω . The main technical innovation is the development of iterated ultrapowers using infinite supports; this requires an application of infinite-exponent partition relations. All our theorems can be proven from hypotheses weaker than AD
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DOI 10.2307/2274328
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Model Theory.Michael Makkai, C. C. Chang & H. J. Keisler - 1991 - Journal of Symbolic Logic 56 (3):1096.
Researches Into the World of "X" [Implies] "X".J. M. Henle - 1979 - Annals of Mathematical Logic 17 (1/2):151.

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