John Baldwin : `Completeness' for Non-Elementary Classes

Posted by , part of the Logic Seminar.

At

Feb. 8, 2011, 3 p.m.

In

SEO 612

Abstract

Much of first order model theory studies the countable models of a complete first order theory. When the sentence is taken to be in $L_{\omega_1,\omega}$, `completeness' (in the sense of proving or disproving every $L_{\omega_1,\omega}$-sense) implies $\aleph_0$-categoricity. So while the Vaught conjecture for complete first order theories has major results, the Vaught conjecture for complete $L_{\omega_1,\omega}$ sentences is trivial. We study a related question. Must a `complete' $\aleph_1$-categorical sentence have at most countably many countable models? We provide several examples where the answer is no under various notions of complete (all of course weaker than above). These notions are attempts at a more semantic definition of complete for AEC.