## Journal of Symbolic Logic

### 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 $\aleph_0$-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 $\aleph_1$-categorical but not $\aleph_0$-categorical saturated $\Sigma^0_1$-structure with a unique computable isomorphism type. In addition, using the construction we give an example of an $\aleph_1$-categorical but not $\aleph_0$-categorical theory whose only non-computable model is the prime one.

#### Article information

Source
J. Symbolic Logic, Volume 72, Issue 3 (2007), 1041-1054.

Dates
First available in Project Euclid: 2 October 2007

https://projecteuclid.org/euclid.jsl/1191333855

Digital Object Identifier
doi:10.2178/jsl/1191333855

Mathematical Reviews number (MathSciNet)
MR2354914

Zentralblatt MATH identifier
1127.03031

#### Citation

Khoussainov, B.; Semukhin, P.; Stephan, F. Applications of Kolmogorov complexity to computable model theory. J. Symbolic Logic 72 (2007), no. 3, 1041--1054. doi:10.2178/jsl/1191333855. https://projecteuclid.org/euclid.jsl/1191333855