Abstract
D’Aquino et al. (J Symb Log 75(1):1–11, 2010) have recently shown that every real-closed field with an integer part satisfying the arithmetic theory IΣ4 is recursively saturated, and that this theorem fails if IΣ4 is replaced by IΔ0. We prove that the theorem holds if IΣ4 is replaced by weak subtheories of Buss’ bounded arithmetic: PV or \({\Sigma^b_1-IND^{|x|_k}}\). It also holds for IΔ0 (and even its subtheory IE 2) under a rather mild assumption on cofinality. On the other hand, it fails for the extension of IOpen by an axiom expressing the Bézout property, even under the same assumption on cofinality.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Boughattas S., Kołodziejczyk L.A.: The strength of sharply bounded induction requires MSP. Ann. Pure Appl. Log. 161(4), 504–510 (2010)
Berarducci A., Otero M.: A recursive nonstandard model of normal open induction. J. Symb. Log. 61(4), 1228–1241 (1996)
Basu, S., Pollack, R., Roy, M.-F.: Algorithms in real algebraic geometry, 2nd edn. Algorithms and Computation in Mathematics, vol. 10. Springer, Berlin (2006)
Boughattas S., Ressayre J.-P.: Bootstrapping, part I. Ann. Pure Appl. Log. 161(4), 511–533 (2010)
Barwise J., Schlipf J.: An introduction to recursively saturated and resplendent models. J. Symb. Log. 41(2), 531–536 (1976)
Buss, S.R.: Bounded arithmetic. Revision of 1985 Princeton University Ph.D. thesis, Bibliopolis, Naples (1986)
Carl, M., D’Aquino, P., Kuhlmann, S.: Real closed exponential fields and models of Peano arithmetic. arXiv:1205.2254 [math.LO] (2012) (Preprint)
Cegielski, P., McAloon, K., Wilmers, G.: Modèles récursivement saturés de l’addition et de la multiplication des entiers naturels, In: van Dalen, D., et al. (eds.) Logic Colloquium ’80, Studies in Logic and the Foundations of Mathematics, vol. 108, pp. 57–68. North-Holland (1982)
Cook S.A., Nguyen P.: Logical Foundations of Proof Complexity. Cambridge University Press, New York (2010)
D’Aquino P., Knight J.F., Starchenko S.: Real closed fields and models of Peano arithmetic. J. Symb. Log. 75(1), 1–11 (2010)
D’Aquino P., Knight J.F., Starchenko S.: Corrigendum to: Real closed fields and models of arithmetic. J. Symb. Log. 77(2), 726 (2012)
Hájek, P., Pudlák, P.: Metamathematics of First-Order Arithmetic, 2nd edn. Perspectives in Mathematical Logic, Springer, Berlin (1998)
Kaye, R.: Open induction, Tennenbaum phenomena, and complexity theory. In: Clote, P., Krajíček, J. (eds.) Arithmetic, Proof Theory, and Computational Complexity, pp. 222–237. Oxford University Press, Oxford (1993)
Krajíček, J.: Bounded Arithmetic, Propositional Logic, and Complexity Theory. Encyclopedia of Mathematics and its Applications, vol. 60. Cambridge University Press, Cambridge (1995)
Marker, D.: Model Theory: An Introduction. Springer, Berlin (2002)
McAloon K.: On the complexity of models of arithmetic. J. Symb. Log. 47(2), 403–415 (1982)
Macintyre A., Marker D.: Primes and their residue rings in models of open induction. Ann. Pure Appl. Log. 43(1), 57–77 (1989)
Mohsenipour, S.: A recursive nonstandard model for open induction with GCD property and confinal primes. In: Enayat, A., et al.(eds.) Logic in Tehran, Lecture Notes in Logic, no. 26, Association for Symbolic Logic, pp. 227–238 (2006)
Mourgues M.-H., Ressayre J.-P.: Every real closed field has an integer part. J. Symb. Log. 58(2), 641–647 (1993)
Marker, D., Steinhorn, C.: Uncountable real closed fields with PA integer parts. arXiv:1205.5156 [math.LO] (2012) (Preprint)
Paris J.B.: O struktuře modelů omezené E 1-indukce. Časopis pro pěstování 109(4), 372–379 (1984)
Shepherdson J.C.: A non-standard model for a free variable fragment of number theory. Bulletin de l’Académie Polonaise des Sciences. Série des Sciences Mathématiques, Astronomiques et Physiques 12(2), 79–86 (1964)
Smith S.T.: Building discretely ordered Bezout domains and GCD domains. J. Algebra 159(1), 191–239 (1993)
Tennenbaum, S.: Non-archimedean models for arithmetic. Not. Am. Math. Soc. 6, 270 (1959)
von zur Gathen, J., Gerhard, J.: Modern Computer Algebra. Cambridge University Press, Cambridge (1999)
Wilkie, A.J.: Some results and problems on weak systems of arithmetic. In: Macintyre, A., et al. (eds.) Logic Colloquium ’77, Studies in Logic and the Foundations of Mathematics, vol. 96, pp. 285–296. North-Holland, (1978)
Wilmers G.: Bounded existential induction. J. Symb. Log. 50(1), 72–90 (1985)
Author information
Authors and Affiliations
Corresponding author
Additional information
Emil Jeřábek supported by RVO: 67985840, grant IAA100190902 of GA AV ČR, project 1M0545 of MŠMT ČR, and a grant from the John Templeton Foundation.
Leszek Aleksander Koł odziejczyk partially supported by grant N N201 382234 of the Polish Ministry of Science and Higher Education.
Rights and permissions
About this article
Cite this article
Jeřábek, E., Kołodziejczyk, L.A. Real closures of models of weak arithmetic. Arch. Math. Logic 52, 143–157 (2013). https://doi.org/10.1007/s00153-012-0311-x
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00153-012-0311-x