Journal of Symbolic Logic

Model Theory Under the Axiom of Determinateness

Mitchell Spector

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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, $\mathscr{L}_{\omega_1\omega}$ is no more powerful than first-order logic. The emphasis then turns to upward Lowenhein-Skolem theorems; $\aleph_1$ is the Hanf number of first-order logic, of $\mathscr{L}_{\omega_1\omega}$, and of a strong fragment of $\mathscr{L}_{\omega_1\omega}$. 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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J. Symbolic Logic, Volume 50, Issue 3 (1985), 773-780.

First available in Project Euclid: 6 July 2007

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Spector, Mitchell. Model Theory Under the Axiom of Determinateness. J. Symbolic Logic 50 (1985), no. 3, 773--780.

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