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A computable ℵ0-categorical structure whose theory computes true arithmetic

Published online by Cambridge University Press:  12 March 2014

Bakhadyr Khoussainov  and
Antonio Montalbán
Bakhadyr Khoussainov
Affiliation:
Department of Computer Science, Auckland University, Auckland, New Zealand. E-mail: bmk@cs.auckland.ac.nz
Antonio Montalbán
Affiliation:
Department of Mathematics, University of Chicago5734 S. University Ave. Chicago, Il 60637, USA. E-mail: antonio@math.uchicago.edu, URL: www.math.uchicago.edu/~antonio
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Abstract

We construct a computable ℵ0-categorical structure whose first order theory is computably equivalent to the true first order theory of arithmetic.

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Type
Research Article
Information
The Journal of Symbolic Logic , Volume 75 , Issue 2 , June 2010 , pp. 728 - 740
Copyright
Copyright © Association for Symbolic Logic 2010

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References

REFERENCES

[AK00]Ash, C. J. and Knight, J., Computable structures and the hyperarithmetical hierarchy, Studies in Logic and the Foundations of Mathematics, vol. 144, North-Holland, Amsterdam, 2000.Google Scholar
[Ers73]Ershov, Ju. L., Skolem functions and constructive models, Algebra i Logika, vol. 12 (1973), pp. 644–654 and 735.Google Scholar
[Gon78]Goncharov, S. S., Strong constructivizability of homogeneous models. Algebra i Logika, vol. 17 (1978), no. 4, pp. 363–388 and 490.Google Scholar
[GHL+03]Goncharov, S. S., Harizanov, V., Laskowski, M., Lempp, S., and Mccoy, C. F., Trivial, strongly minimal theories are model complete after naming constants, Proceedings of the American Mathematical Society, vol. 131 (2003), no. 12, pp. 3901–3912, (electronic).CrossRefGoogle Scholar
[GK04]Goncharov, S. S. and Khoussainov, B., Complexity of theories of computable categorical models. Algebra i Logika, vol. 43 (2004), no. 6, pp. 650–665 and 758–759.Google Scholar
[GN73]Goncharov, S. S. and Nurtazin, A. T., Constructive models of complete decidable theories, Algebra i Logika, vol. 12 (1973), pp. 125–142 and 243.Google Scholar
[Har74]Harrington, L., Recursively presentable prime models, this Journal, vol. 39 (1974), pp. 305–309.Google Scholar
[Hod93]Hodges, W., Model theory. Encyclopedia of Mathematics and its Applications, vol. 42, Cambridge University Press, Cambridge, 1993.CrossRefGoogle Scholar
[Khi98]Khisamiev, N. G., Constructive abelian groups. Handbook of recursive mathematics, volume 2, Studies in Logic and the Foundations of Mathematics, vol. 139, North-Holland, Amsterdam, 1998, pp. 1177–1231.Google Scholar
[KLLS07]Khoussainov, B., Laskowski, M., Lempp, S., and Solomon, R., On the computability-theoretic complexity of trivial, strongly minimal models, Proceedings of the American Mathematical Society, vol. 135 (2007), no. 11, pp. 3711–3721, (electronic).CrossRefGoogle Scholar
[KNS97]Khoussainov, B., Nies, A., and Shore, R. A., Computable models of theories with few models, Notre Dame Journal of Formal Logic, vol. 38 (1997), no. 2. pp. 165–178.CrossRefGoogle Scholar
[Kni94]Knight, J., Nonarithmetical ℵ0-categorical theories with recursive models, this Journal, vol. 59 (1994), no. 1, pp. 106–112.Google Scholar
[Kud80]Kudaibergenov, K. Z., Constructivizable models of undecidable theories, Sibirskiĭ Matematicheskiĭ Zhurnal, vol. 21 (1980), no. 5, pp. 155–158 and 192.Google Scholar
[LS79]Lerman, M. and Schmerl, J. H., Theories with recursive models, this Journal, vol. 44 (1979), no. 1, pp. 59–76.Google Scholar
[Mil78]Millar, T. S., Foundations of recursive model theory, Annals of Mathematical Logic, vol. 13 (1978), no. 1, pp. 45–72.CrossRefGoogle Scholar
[Mil81]Millar, T. S., Vaught's theorem recursively revisited, this Journal, vol. 46 (1981), no. 2, pp. 397–411.Google Scholar
[Mor76]Morley, M., Decidable models, Israel Journal of Mathematics, vol. 25 (1976), no. 3–4, pp. 233–240.CrossRefGoogle Scholar
[Per78]Peretjatkin, M. G., A criterion of strong constructivizability of a homogeneous model, Algebra i Logika, vol. 17 (1978), no. 4, pp. 436–454.Google Scholar
[Sch78]Schmerl, J. H., A decidable ℵ0-categorical theory with a nonrecursive Ryll–Nardzewski function. Fundamenta Mathematicae, vol. 98 (1978), no. 2, pp. 121–125.CrossRefGoogle Scholar