ON THE NUMBER OF MODELS OF UNCOUNTABLE THEORIES

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) greater-than-or-equal-to N0. (ii) If T is not unidimensional, then for any lambda greater-than-or-equal-to \T\, I(lambda, T) greater-than-or-equal-to N0. Theorem B. Let T be superstable, not totally transcendental and nonmultidimensional. Let theta(x) be a formula of least R(infinity) rank which does not have Morley rank, and let p be any stationary completion of theta which also fails to have Morley rank. Then p is regular and locally modular.