Hostname: page-component-8448b6f56d-c47g7 Total loading time: 0 Render date: 2024-04-19T20:03:50.820Z Has data issue: false hasContentIssue false
  • English
  • Français

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
Get access Rights & Permissions [Opens in a new window]

Abstract

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

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

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