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On the number of models of uncountable theories

Published online by Cambridge University Press:  12 March 2014

Ambar Chowdhury  and
Anand Pillay
Ambar Chowdhury
Affiliation:
Department of Mathematics and Statistics, McMaster University, Hamilton, Ontario, Canada, L8S 4K1, E-mail: achowdhu@johnny.math.mcmaster.ca
Anand Pillay
Affiliation:
Department of Mathematics, University of Notre Dame, Notre Dame, Indiana 46556, E-mail: anand.pillay.l@nd.edu
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Abstract

In this paper we establish the following theorems.

Theorem A. Let T be a complete first-order theory which is uncountable, Then:

(i) I(∣T∣, T) ≥ ℵ0

(ii) If T is not unidimensional, then for any λ ≥ ∣T∣, I(λ, T) ≥ ℵ0.

Theorem B. Let T be superstable, not totally transcendental and nonmultidimensional. Let θ(x) be a formula of least R∞ rank which does not have Morley rank, and let p be any stationary completion of θ which also fails to have Morley rank. Then p is regular and locally modular.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 59 , Issue 4 , December 1994 , pp. 1285 - 1300
Copyright
Copyright © Association for Symbolic Logic 1994

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