SPEAKER: Andre Nies (University of Chicago)

TITLE: Spectra of $\omega_1$-categorical theories

ABSTRACT: If $T$ is an $\omega_1$ categorical but not $\omega$-categorical theory, then the countable models of $T$ form an elementary chain

$$\A_0 \prec \A_1 \prec \ \ldots \ \prec \A_\omega.$$

Let $SRM(T)$ be the set of those $i \le \omega$ such that $\A_i$ has a recursive presentation (SRM stands for ``spectrum of recursive models''). While it can be shown that $SRM(T)$ is recursive in $\emptyset^{(\omega+4)}$, it is not known which sets can occur as sets $SRM(T)$. Extending results by Khoussainov, Nies and Shore we obtain examples of theories $T$ such that $SRM(T) = [1, \nu)$, for any $\nu$ such that $1 < \nu \le \omega$. These seem to be the most complicated spectra obtained so far.