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Answer to a problem raised by J. Robinson: The arithmetic of positive or negative integers is definable from successor and divisibility

Published online by Cambridge University Press:  12 March 2014

Denis Richard
Denis Richard*
Affiliation:
Département de Mathématiques, Université Claude Bernard—Lyon 1, 69622 Villeurbanne Cédex, France
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Abstract

In this paper we give a positive answer to Julia Robinson's question whether the definability of + and · from S and ∣ that she proved in the case of positive integers is extendible to arbitrary integers (cf. [JR, p. 102]).

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 50 , Issue 4 , December 1985 , pp. 927 - 935
Copyright
Copyright © Association for Symbolic Logic 1985

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References

REFERENCES

[AS]Schinzel, A., On primitive prime factors of an – bn, Proceedings of the Cambridge Philosophical Society, vol. 58 (1962), pp. 555–562.CrossRefGoogle Scholar
[AW]Woods, A., Some problems in logic and number theory and their connections, Thesis, University of Manchester, Manchester, 1981, pp. 51–70, 121–122.Google Scholar
[BV]Birkhoff, G. D. and Vandiver, H. S., On the integral divisors of an – bn, Annals of Mathematics, ser. 2, vol. 5 (1904), pp. 173–180.CrossRefGoogle Scholar
[DR1]Richard, D., La théorie sans égalité du successeur et de la coprimarité des entiers naturels est indécidable. Le prédicat de primarité est définissable dans le langage de cette théorie, Comptes Rendus des Séances de l'Académie des Sciences. Série I: Mathématique, vol. 294 (1982), pp. 143–146.Google Scholar
[DR2]Richard, D., The arithmetics as theories of two orders, Orders: description and roles (Pouzet, M. and Richard, D., editors), Annals of Discrete Mathematics, vol. 23, North-Holland, Amsterdam, 1984, pp. 287–311.Google Scholar
[DR3]Richard, D., All arithmetical sets of powers of primes are first order definable in terms of the successor function and the coprimeness predicate, Discrete Mathematics (to appear in the special issue devoted to the proceedings of the conference on ordered sets and their applications, Lyon, 1982).Google Scholar
[DR4]Richard, D., Definability problems in terms of successor and coprimeness over arbitrary integers, Preprint.Google Scholar
[HE]Enderton, H. B., A mathematical introduction to logic, Academic Press, New York, 1970.Google Scholar
[JR]Robinson, J., Definability and decision problems in arithmetic, this Journal, vol. 14 (1949), pp. 98–114.Google Scholar
[KZ]Zsigmondy, K., Zur Theorie der Potenzreste, Monatshefte für Mathematik und Physik, vol. 3 (1892), pp. 265–284.CrossRefGoogle Scholar