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.