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EXISTENTIALLY CLOSED MODELS IN THE FRAMEWORK OF ARITHMETIC

Published online by Cambridge University Press:  10 May 2016

ZOFIA ADAMOWICZ ,
ANDRÉS CORDÓN-FRANCO  and
F. FÉLIX LARA-MARTÍN
ZOFIA ADAMOWICZ
Affiliation:
MATHEMATICAL INSTITUTE OF THE POLISH ACADEMY OF SCIENCES ŚNIADECKICH 8 00-956 WARSZAWA, POLANDE-mail:[email protected]
ANDRÉS CORDÓN-FRANCO
Affiliation:
DEPARTMENT OF COMPUTER SCIENCE AND ARTIFICIAL INTELLIGENCE UNIVERSIDAD DE SEVILLA AVDA. REINA MERCEDES S/N 41012 SEVILLA, SPAINE-mail:[email protected]
F. FÉLIX LARA-MARTÍN
Affiliation:
DEPARTMENT OF COMPUTER SCIENCE AND ARTIFICIAL INTELLIGENCE UNIVERSIDAD DE SEVILLA AVDA. REINA MERCEDES S/N 41012 SEVILLA, SPAINE-mail:[email protected]
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Abstract

We prove that the standard cut is definable in each existentially closed model of IΔ0 + exp by a (parameter free) П1–formula. This definition is optimal with respect to quantifier complexity and allows us to improve some previously known results on existentially closed models of fragments of arithmetic.

Keywords

fragments of Peano arithmeticExistentially closed modelsTuring degrees of arithmetic theories
Type
Articles
Information
The Journal of Symbolic Logic , Volume 81 , Issue 2 , June 2016 , pp. 774 - 788
Copyright
Copyright © The Association for Symbolic Logic 2016 

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References

REFERENCES

Adamowicz, Z., On maximal theories, this journal, vol. 56 (1991), pp. 885–890.Google Scholar
Adamowicz, Z. and Bigorajska, T., Existentially closed structures and Gödel’s second incompleteness theorem, this Journal, vol. 66 (2001), pp. 349–356.Google Scholar
Adamowicz, Z., Kołodziejczyk, L. A., and Paris, J.B., Truth definitions without exponentiation and the Σ1 collection scheme, this Journal, vol. 77 (2012), pp. 649–655.Google Scholar
Cordón-Franco, A., Fernández-Margarit, A., and Lara-Martín, F. F., A note on Σ1-maximal models, this Journal, vol. 72 (2007), pp. 1072–1078.Google Scholar
Goldrei, D. C., Macintyre, A., and Simmons, H., The forcing companions of number theories . Israel Journal of Mathematics, vol. 14 (1973), pp. 317–337.CrossRefGoogle Scholar
Hájek, P. and Pudlák, P., Metamathematics of First-order Arithmetic, Perspectives in Mathematical Logic, Springer–Verlag, Berlin, 1993.Google Scholar
Hirschfeld, J. and Wheeler, W., Forcing, Arithmetic, Division rings, Lecture Notes in Mathematics, vol. 454, Springer–Verlag, Berlin, 1975.Google Scholar
Kaye, R., Models of Peano Arithmetic, Oxford Logic Guides, vol. 15, Oxford University Press, Oxford, 1991.CrossRefGoogle Scholar
Kotlarski, H., An addition to Rosser’s theorem, this Journal, vol. 61 (1996), pp. 285–292.Google Scholar
Lessan, H., Models of Arithmetic , Ph.D. thesis, University of Manchester, 1978.Google Scholar
Macintyre, A. and Simmons, H., Algebraic properties of number theories . Israel Journal of Mathematics, vol. 22 (1975), pp. 7–27.Google Scholar
McAloon, K., Completeness theorems, incompleteness theorems and models of arithmetic . Transactions of the American Mathematical Society, vol. 239 (1978), pp. 253–277.CrossRefGoogle Scholar
Odifreddi, P. G., Classical Recursion Theory, Studies in Logic and the Foundations of Mathematics, vol. 125, North-Holland, Amsterdam, 1989.Google Scholar
Robinson, A., Non standard arithmetic and generic arithmetic , Logic, Methodology and Philosophy of Science IV (Suppes, P. et al. , editors), North-Holland, Amsterdam, 1974, pp. 137–154.Google Scholar
Simmons, H., Existentially closed structures, this Journal, vol. (1972), pp. 293–310.Google Scholar
Simmons, H., Existentially closed models of basic number theory , Logic Colloquium 76 (Gandy, R. and Hyland, M., editors), Studies in Logic and the Foundations of Mathematics, vol. 87, North-Holland, Amsterdam, 1977, pp. 325–369.Google Scholar
Wilkie, A. J. and Paris, J. B., On the scheme of induction for bounded arithmetic formulas . Annals of Pure and Applied Logic, vol. 35 (1987), pp. 261–302.CrossRefGoogle Scholar