On Ω 1 -categorical but Not Ω-categorical Theories

1 ABSTRACT By generalizing to model theory notions connected with dimension, it is shown that certain theories categorical in uncountable powers have countably many de-numerable isomorphism types which are arranged in an ω + 1 sequence under the ordering of the possibility of elementary imbedding. It is also shown that any countable elementary extension of a denumerable saturated model of a theory categorical in uncountable powers is saturated. PREFACE This thesis should be readable by anyone with a little background in logic; an excellent place to get such a background is the expository paper [7] of R. L. Vaught. Professor Vaught pointed out an error in a preliminary version of this paper and made several useful suggestions about how to proceed. Professor M. Morley in letters and conversation was most helpful and, in particular, pointed out to me a theorem which is implicit in [3] that turned out to be the key to applying the results of Chapter 1 of this thesis to theories categorical in ω 1 but not ω. I would like to take this opportunity to thank both of these men. I would like to thank my advisor, Professor Donald Kreider, who was more than generous with his time and help.