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Applications of Kolmogorov complexity to computable model theory

Published online by Cambridge University Press:  12 March 2014

Bakhadyr Khoussainov ,
Pavel Semukhin  and
Frank Stephan
Bakhadyr Khoussainov
Affiliation:
Department of Computer Science, University of Auckland, Private Bag 92019, Auckland, New Zealand, E-mail: bmk@cs.auckland.ac.nz
Pavel Semukhin
Affiliation:
Department of Computer Science, University of Auckland, Private Bag 92019, Auckland, New Zealand, E-mail: pavel@cs.auckland.ac.nz
Frank Stephan
Affiliation:
National University of Singapore, Mathematical Institute, 2 Science Drive 2, Singapore117543, Republic of Singapore, E-mail:, fstephan@comp.nus.edu.sg
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Abstract

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 isomor-phism 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 -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
Research Article
Information
The Journal of Symbolic Logic , Volume 72 , Issue 3 , September 2007 , pp. 1041 - 1054
Copyright
Copyright © Association for Symbolic Logic 2007

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