Series: Penn State Logic Seminar Date: Tuesday, November 25, 2003 Time: 2:30 - 3:45 PM Place: 113 McAllister Building Speaker: Carl Mummert, Penn State University, Mathematics Title: An Incompleteness Theorem for beta_n-Models Abstract: Let omega denote the set of natural numbers, and P(omega) the powerset of omega. For n a positive integer, a beta_n-model is a subset of P(omega) which is a Sigma^1_n-elementary submodel of P(omega). In this talk I will discuss recent joint work with Stephen Simpson. The main result is a beta_n-model version of G"odel's Second Incompleteness Theorem: if a recursively axiomatized theory T has a beta_n-model, then so does T + ``there is no countable beta_n-model of T.'' I will discuss several corollaries of this theorem, including (1) the existence of a beta_n model which is not a beta_{n+1} model, (2) a beta_n-model version of L"ob's Theorem. This talk should be accessible to graduate students who have taken a course in mathematical logic.