H. Jerome Keisler: Randomizations of scattered theories

Place: Bahen Centre (Room BA6183)

Date: 30-June-2015 (14:00-15:30)

Speaker: H. Jerome Keisler.
Title: Randomizations of scattered theories

Abstract: Consider a sentence $\phi$ of the infinitary logic $L_{\omega_1, \omega}$. In 1970, Morley introduced the notion of a scattered sentence, and showed that if $\phi$ is scattered then the class $I(\phi)$ of isomorphism types of countable models of $\phi$ has cardinality at most $\aleph_1$, and if $\phi$ is not scattered then $I(\phi)$ has cardinality continuum. The absolute form of Vaught’s conjecture for $\phi$ says that if $\phi$ is scattered then $I(\phi)$ is at most countable. Generalizing previous work of Ben Yaacov and the author, we introduce here the notion of a separable model of $\phi^R$, which is a separable continuous structure whose elements are random elements of a model of $\phi$. We say that $\phi^R$ has few separable models if every separable model of $\phi^R$ is uniquely characterized up to isomorphism by a function that assigns probabilities summing to one to countably many elements of $I(\phi)$. In a previous paper, Andrews and the author showed that if $\phi$ is a complete first order theory and $I(\phi)$ is at most countable then $\phi^R$ has few separable models. We show here that this result holds for all $\phi$, and that if $\phi^R$ has few separable models then $\phi$ is scattered. Hence if the absolute Vaught conjecture holds for $\phi$, then $\phi^R$ has few separable models if and only if $I(\phi)$ is countable, and also if and only if $\phi$ is scattered. Moreover, assuming Martin’s axiom for $\aleph_1$, we show that if $\phi$ is scattered then $\phi^R$ has few separable models.

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