## 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

#### Citation

Spector, Mitchell. Model Theory Under the Axiom of Determinateness. J. Symbolic Logic 50 (1985), no. 3, 773--780. https://projecteuclid.org/euclid.jsl/1183741911