Applications of Kolmogorov complexity to computable model theory

In this paper we answer the following well-known open question in computable model theory. Does there exist a computable not ℵ0-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 ℵ1-categorical but not ℵ0-categorical saturated Σ0 1-structure with a unique computable isomorphism type. In addition, using the construction we give an example of an ℵ1-categorical but not ℵ0-categorical theory whose only non-computable model is the prime one.

• Type: Journal
• Year: 2007
• Where: JSYML
• Authors: Bakhadyr Khoussainov, Pavel Semukhin, Frank Stephan
• Keywords: Computable Isomorphism Type, Computable Model Theory, JSYML 2007, Unique Computable Isomorphism