An Incompleteness Theorem for beta_n-Models

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.