Categoricity without equality

We study categoricity in power for reduced models of rst order logic without equality. 1. Introduction. The object of this paper is to study categoricity in power for theories in rst order logic without equality. Our results will re- veal some surprising dierences between the model theory for logic without equality and for logic with equality. When we consider categoricity, it is natural to identify elements which are indistinguishable from each other. We will do this by conning our attention to reduced models, that is, modelsM such that any pair of elements which satisfy the same formulas with parameters inM are equal. We also conne our attention to complete theories T in a countable language such that all models of T are innite. T is said to be -categorical if T has exactly one reduced model of cardinality up to isomorphism. The classical result about !-categoricity for logic with equality is the Ryll-Nardzewski theorem, which says that T is !-categorical if and only if T has only nitely many completen-types for each nite n. This result fails for logic without equality. Another relevant result which fails for logic without equality is the Lowenheim{Skolem{Tarski theorem, that T has at least one model of every innite cardinality. Concerning uncountable categoricity, ao- (a) conjectured that if T is -categorical for some uncountable , then T is -categorical for every uncountable . The ao- conjecture was proved for logic with equality by Morley (M). We will show that this result also holds for logic without equality.