On the number of models of uncountable theories

Abstract

AbstractIn 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.