On the Strong Martin Conjecture

Journal of Symbolic Logic 56 (3):862-875 (1991)

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 $L_1$, 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.

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The Trend of Logic and Foundation of Mathematics in Japan in 1991 to 1996.Yuzuru Kakuda, Kanji Namba & Nobuyoshi Motohashi - 1997 - Annals of the Japan Association for Philosophy of Science 9 (2):95-110.

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