Logic Seminar
John Baldwin
UIC
Complexity and Absoluteness in $L_{\omega_1,\omega}$
Abstract: Translation from a sentence of $L_{\omega_1,\omega}$ to an associated atomic class is a key tool for the
study of categoricity in infinitary logic.
We compare the complexity of definition of model theoretic notions such as $\omega$-stability and
excellence for a sentence of $L_{\omega_1,\omega}$ and the associated atomic class. We show these properties are $\Sigma^1_2$ on the sentences and $\Pi^1_1$ on the classes. (Lower bounds have not been established in the atomic class case). In either case these properties are absolute. But the question of whether $\aleph_1$-categoricity is absolute remains open for either formulation. Connecting this study with more classical descriptive set theory we show that the class of models of a sentence of $L_{\omega_1,\omega}$ whose automorphism groups admit a complete left invariant metric is $\Pi^1_1$ but not $\Sigma^1_1$. Techniques from model theory, recursion theory and descriptive set theory are used. Much of this is joint work with David Marker.
Tuesday October 12, 2010 at 4:00 PM in SEO 612