The Structure of Models of Uncountably Categorical Theories

The natural notion of categoricity, as it was discovered in the 1930's, is degenerate for first order languages, since only a finite structure can be described up to isomorphism by its first order theory. This has led to a new notion of categoricity. A theory is said to be categorical in a power if it has a model of this power which is unique up to isomorphism. Morley has proved, answering the well-known question of îioê, that for the spectrum of the infinite powers in which a given first order theory is categorical, the only possibilities are: (i) all infinite powers, (ii) all uncountable powers, (iii) «o, (iv) empty. Theories having (i) or (ii) as the categoricity spectrum are called uncountably categorical. Totally categorical theories are those corresponding to (i) only. The same terms are used for models of such theories. The countable N0-categorical structures are characterized by the Byll-Nardzewski-Svenonius-Engeler theorem as those structures M for which there are only finitely many orbits under the action of Aut M on JH, for every n. This is in fact the only general theorem known in this case and there are no grounds for expecting a good general theory. We have an entirely different situation in the uncountably categorical case. Natural examples here are an algebraically closed field, a simple algebraic group over an algebraically closed field [19], a vector space