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EXISTENTIALLY CLOSED MODELS IN THE FRAMEWORK OF ARITHMETIC

  • ZOFIA ADAMOWICZ (a1), ANDRÉS CORDÓN-FRANCO (a2) and F. FÉLIX LARA-MARTÍN (a3)
Abstract
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.

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D. C. Goldrei , A. Macintyre , and H. Simmons , The forcing companions of number theories. Israel Journal of Mathematics, vol. 14 (1973), pp. 317337.

P. Hájek and P. Pudlák , Metamathematics of First-order Arithmetic, Perspectives in Mathematical Logic, Springer–Verlag, Berlin, 1993.

J. Hirschfeld and W. Wheeler , Forcing, Arithmetic, Division rings, Lecture Notes in Mathematics, vol. 454, Springer–Verlag, Berlin, 1975.

A. Macintyre and H. Simmons , Algebraic properties of number theories. Israel Journal of Mathematics, vol. 22 (1975), pp. 727.

K. McAloon , Completeness theorems, incompleteness theorems and models of arithmetic. Transactions of the American Mathematical Society, vol. 239 (1978), pp. 253277.

A. J. Wilkie and J. B. Paris , On the scheme of induction for bounded arithmetic formulas. Annals of Pure and Applied Logic, vol. 35 (1987), pp. 261302.

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The Journal of Symbolic Logic
  • ISSN: 0022-4812
  • EISSN: 1943-5886
  • URL: /core/journals/journal-of-symbolic-logic
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