[FOM] necessary conditions for obtaining uncountable models: Gregory's theorem

[John Baldwin]

J. Johnson, J. F. Knight, V. Ocasio, and S. VanDenDriessche at Notre Dame 
have recently given an example showing the assumption that T is Sigma_1 on 
A is essential in the theorem below.


Theorem 1 (Gregory 1970).  If A is a countable admissible set and T is an 
L_A-theory that is Sigma _1 on A, then T has an uncountable model iff and only 
if  it has a model with a proper L_A-elementary extension.

Gregory had announced the existence of such an example but apparently did 
not publish it.

Does anyone know of other examples than the one obtained 
by the Notre Dame group?




John T. Baldwin
Professor Emeritus
Department of Mathematics, Statistics,
and Computer Science M/C 249
jbaldwin at uic.edu
851 S. Morgan
Chicago IL
60607