Model Theory seminar: A counterexample to the infinitary Vaught conjecture

The beginning of modern model theory can be traced back to a paper of Vaught titled “Denumerable models of complete theories” in 1961. At the end of that paper Vaught asked the following question which gave rise to the conjecture that bears his name:  Can it be proved, without the use of the continuum hypothesis, that there exists a complete theory having exactly $\aleph_1$ non-isomorphic denumerable models?   In the first part of the talk, I will review background material on Vaught’s conjecture including basic results in $L_{\omega_1, \omega}$ as well as the connections with descriptive set theory.  In the second part, I will present a counterexample to the infinitary Vaught conjecture.  This is joint work with Cameron Freer and Robin Knight.   No logic other than what could be found in an introductory undergraduate course will be assumed.