Logic Seminar
John Baldwin
UIC
`Completeness' for Non-Elementary Classes
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.
Tuesday February 8, 2011 at 3:00 PM in SEO 612