Semi-minimal theories and categoricity

Abstract

The study of countable theories categorical in some uncountable power was initiated by Łoś and Vaught and developed in two stages. First, Morley proved (1962) that a countable theory categorical in some uncountable power is categorical in every uncountable power, a conjecture of Łoś. Second, Baldwin and Lachlan confirmed (1969) Vaught's conjecture that a countable theory categorical in some uncountable power has either one or countably many isomorphism types of countable models. That result was obtained by pursuing a line of research developed by Marsh (1966). For certain well-behaved theories, which he called strongly minimal, Marsh's method yielded a simple proof of Łoś's conjecture and settled Vaught's conjecture.In recent years efforts have been made to extend these results to uncountable theories. The generalized Łoś conjecture states that a theory T categorical in some power greater than ∣T∣ is categorical in every such power. It was settled by Shelah (1970). Shelah then raised the question of the models in power ∣T∣ = ℵα of a theory T categorical in ∣T∣+, conjecturing in [S3] that there are exactly ∣α∣ + ℵ0 such models, up to isomorphism. This conjecture provided the initial motivation for the present work. We define and study semi-minimal theories analogous in some ways to Marsh's strongly minimal (countable) theories. We describe the models of a semi-minimal theory T which contain an infinite indiscernible set. Besides throwing some light on Shelah's conjecture, our method gives simple proofs of the Łoś conjecture and of the Morley conjecture on categoricity in ∣T∣, in the case of a semi-minimal theory T. Other results as well as some examples are provided.