Constructive Models of Uncountably Categorical Theories

We construct a strongly minimal (and thus uncountably categorical) but not totally categorical theory in a nite language of binary predicates whose only constructive (or recursive) model is the prime model. 0. Introduction. E ective (or recursive) model theory studies to which degree constructions in model theory and algebra can be made e ective. A presentation of a countable model M is an isomorphic copy N with universe N = !. An e ective (or computable, or recursive) presentation is one where all the relations, functions, and constants on N are given by uniformly computable functions. Now, for a countable model M of a rst-order theory T , there are various degrees to which the construction of M can be made e ective: We call the model M constructive (or recursive, or computable) if it has an e ective presentation, or equivalently if its open diagram (i.e., the collection of all quanti er-free sentences true in (M; a) a2M 1991 Mathematics Subject Classi cation. 03C57, 03D45.