Applications of kolmogorov complexity to computable model theory

Abstract

In this paper we answer the following well-known open question in computable model theory. Does there exist a computable not N0- categorical saturated structure with a unique computable isomorphism type? Our answer is affirmative and uses a construction based on Kolmogorov complexity. With a variation of this construction, we also provide an example of an N 1 -categorical but not N0-categorical saturated Σ1 0-structure with a unique computable isomorphism type. In addition, using the construction we give an example of an N 1-categorical but not N0-categorical theory whose only non-computable model is the prime one. © 2007, Association for Symbolic Logic.