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Real closed fields and models of Peano arithmetic

Published online by Cambridge University Press:  12 March 2014

P. D'Aquino ,
J. F. Knight  and
S. Starchenko
P. D'Aquino
Affiliation:
Dipartimento di Matematica, Seconda Università di NapoliVia Vivaldi 43, Caserta 81100, Italy, E-mail: paola.daquino@unina2.it
J. F. Knight
Affiliation:
Department of Mathematics, University of Notre Dame, 255 Hurley Hall, Notre Dame, Indiana 46556, USA, E-mail: Julia.F.Knight.l@nd.edu
S. Starchenko
Affiliation:
Department of Mathematics, University of Notre Dame, 255 Hurley Hall, Notre Dame, Indiana 46556, USA, E-mail: sstarche@nd.edu
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Abstract

Shepherdson [14] showed that for a discrete ordered ring I, I is a model of I Open iff I is an integer part of a real closed ordered field. In this paper, we consider integer parts satisfying PA. We show that if a real closed ordered field R has an integer part I that is a nonstandard model of PA (or even IΣ4), then R must be recursively saturated. In particular, the real closure of I, RC (I), is recursively saturated. We also show that if R is a countable recursively saturated real closed ordered field, then there is an integer part I such that R = RC(I) and I is a nonstandard model of PA.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 75 , Issue 1 , March 2010 , pp. 1 - 11
Copyright
Copyright © Association for Symbolic Logic 2012

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References

REFERENCES

[1] Barwise, J. and Schlipf, J., An introduction to recursively saturated and resplendent models, this Journal, vol. 41 (1976), pp. 531–536.Google Scholar
[2] Biljakovic, D., Kochetov, M., and Kuhlmann, S., Primes and irreducibles in truncation integer parts of real closed fields, Logic, algebra and arithmetic, Lecture Notes in Logic, vol. 26, Association for Symbolic Logic, La Jolla, CA, 2006, pp. 42–65.Google Scholar
[3] Boughattas, S. and Ressayre, J.-P., Arithmetization of the field of reals with exponentiation, RAIRO-Theoretical Informatics and Applications, vol. 42 (2008), pp. 105–119.Google Scholar
[4] Cegielski, P., McAloon, K., and Wilmers, G., Modèles recursivement saturés de l'addition et de la multiplication des entiers naturels, Logic Colloquium '80 (Prague, 1980), North-Holland, Amsterdam, 1982, pp. 57–68.Google Scholar
[5] Jacobson, N., Lectures in abstract algebra. Volume III, Springer-Verlag, 1975.Google Scholar
[6] Knight, J. F., Pillay, A., and Steinhorn, C., Definable sets in ordered structures. II, Transactions of the American Mathematical Society, vol. 295 (1986), pp. 593–605.CrossRefGoogle Scholar
[7] Lipshitz, L. and Nadel, M., The additive structure of models of arithmetic, Proceedings of the American Mathematical Society, vol. 68 (1978), no. 3, pp. 331–336.Google Scholar
[8] Mourgues, M.-H. and Ressayre, J. P., Every real closed field has an integer part, this Journal, vol. 58 (1993), pp. 641–647.Google Scholar
[9] Pillay, A. and Steinhorn, C., Definable sets in ordered structures. I, Transactions of the American Mathematical Society, vol. 295 (1986), pp. 565–592.CrossRefGoogle Scholar
[10] Ressayre, J.-P., Integer parts of real closed exponential fields, Arithmetic, proof theory, and computational complexity (Clote, P. and Krajicek, J., editors), Oxford University Press, New York, 1993.Google Scholar
[11] Schlipf, J., A guide to the identification of admissible sets above structures, Annals of Pure and Applied Logic, vol. 12 (1977), pp. 151–192.Google Scholar
[12] Schmerl, J. H., Recursively saturated models generated by indiscernibles, Notre Dame Journal of Formal Logic, vol. 26 (1985), pp. 99–105.Google Scholar
[13] Scott, D., Algebras of sets binumerable in complete extensions of arithmetic, Recursive function theory (Dekker, J., editor), American Mathematical Society, 1962, pp. 117–121.Google Scholar
[14] Shepherdson, J., A non-standard model for a free variable fragment of number theory, Bulletin de l'Academic Polonaise des Sciences, vol. 12 (1964), pp. 79–86.Google Scholar