Journal of Symbolic Logic

Model Theory Under the Axiom of Determinateness

Mitchell Spector

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, $\mathscr{L}_{\omega_1\omega}$ 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.

Article information

Source
J. Symbolic Logic, Volume 50, Issue 3 (1985), 773-780.

Dates
First available in Project Euclid: 6 July 2007

https://projecteuclid.org/euclid.jsl/1183741911

Mathematical Reviews number (MathSciNet)
MR805683

Zentralblatt MATH identifier
0588.03019

JSTOR