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 ${\mathrm{\aleph }}_0$-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 ${\mathrm{\aleph }}_1$-categorical but not ${\mathrm{\aleph }}_0$-categorical saturated ${\mathrm{\Sigma }}_1^0$-structure with a unique computable isomorphism type. In addition, using the construction we give an example of an ${\mathrm{\aleph }}_1$-categorical but not ${\mathrm{\aleph }}_0$-categorical theory whose only non-computable model is the prime one.