## Journal of Symbolic Logic

### On the Strong Martin Conjecture

Masanori Itai

#### Abstract

We study the following conjecture. Conjecture. Let $T$ be an $\omega$-stable theory with continuum many countable models. Then either i) $T$ has continuum many complete extensions in $L_1(T)$, or ii) some complete extension of $T$ in $L_1$ has continuum many $L_1$-types without parameters. By Shelah's proof of Vaught's conjecture for $\omega$-stable theories, we know that there are seven types of $\omega$-stable theory with continuum many countable models. We show that the conjecture is true for all but one of these seven cases. In the last case we show the existence of continuum many $L_2$-types.

#### Article information

Source
J. Symbolic Logic, Volume 56, Issue 3 (1991), 862-875.

Dates
First available in Project Euclid: 6 July 2007

Permanent link to this document
https://projecteuclid.org/euclid.jsl/1183743734

Digital Object Identifier
doi:10.2178/jsl/1183743734

Mathematical Reviews number (MathSciNet)
MR1129150

Zentralblatt MATH identifier
0743.03022

JSTOR
links.jstor.org

#### Citation

Itai, Masanori. On the Strong Martin Conjecture. J. Symbolic Logic 56 (1991), no. 3, 862--875. doi:10.2178/jsl/1183743734. https://projecteuclid.org/euclid.jsl/1183743734