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On Diophantine equations solvable in models of open induction

Published online by Cambridge University Press:  12 March 2014

Margarita Otero
Margarita Otero*
Affiliation:
Mathematical Institute, Oxford OX1 3LB, England
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Abstract

We consider IOpen, the subsystem of PA (Peano Arithmetic) with the induction scheme restricted to quantifier-free formulas.

We prove that each model of IOpen can be embedded in a model where the equation has a solution. The main lemma states that there is no polynomial f{x,y) with coefficients in a (nonstandard) DOR M such that ∣f(x,y) ∣ < 1 for every (x,y) Є C, where C is the curve defined on the real closure of M by C: x2 + y2 = a and a > 0 is a nonstandard element of M.

Information

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 55 , Issue 2 , June 1990, pp. 779 - 786
Copyright
Copyright © Association for Symbolic Logic 1990

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References

REFERENCES

[1]Borevich, Z. I. and Shavarevich, I. R., Number theory, Academic Press, New York, 1968.Google Scholar
[2]van den Dries, L., Some model theory and number theory for models of weak systems of arithmetic, Model theory of algebra and arithmetic (Karpacz, 1979; Pacholski, L.et al., editors), Lecture Notes in Mathematics, vol. 834, Springer-Verlag, Berlin, 1980, pp. 346–362.CrossRefGoogle Scholar
[3]Macintyre, A. J. and Marker, D., Primes and their residue rings in models of open induction, Annals of Pure and Applied Logic, vol. 43 (1989), pp. 57–77.CrossRefGoogle Scholar
[4]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, vol. 12 (1964), pp. 79–86.Google Scholar
[5]Wilkie, A. J., Some results and problems on weak systems of arithmetic, Logic Colloquium '77 (Macintyre, A. J.et al., editors), North-Holland, Amsterdam, 1978, pp. 285–296.CrossRefGoogle Scholar