Uncountable theories that are categorical in a higher power

Abstract

AbstractIn this paper we prove three theorems about first-order theories that are categorical in a higher power. The first theorem asserts that such a theory either is totally categorical or there exist prime and minimal models over arbitrary base sets. The second theorem shows that such theories have a natural notion of dimension that determines the models of the theory up to isomorphism. From this we conclude that I(T,ℵα,) = ℵ0 + ∣α∣ where ℵα = the number of formulas modulo T-equivalence provided that T is not totally categorical. The third theorem gives a new characterization of these theories.