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Definability in terms of the successor function and the coprimeness predicate in the set of arbitrary integers

Published online by Cambridge University Press:  12 March 2014

Denis Richard
Denis Richard*
Affiliation:
Institute De Mathématiques Et Informatique, Université Claude Bernard (Lyon 1), 69622 Villeurbanne, France
*
Institut Universitaire de Technologie, Université de Clermont-Ferrand I, 63170 Aubière, France
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Abstract

Using coding devices based on a theorem due to Zsigmondy, Birkhoff and Vandiver, we first define in terms of successor S and coprimeness predicate ⊥ a full arithmetic over the set of powers of some fixed prime, then we define in the same terms a restriction of the exponentiation. Hence we prove the main result insuring that all arithmetical relations and functions over prime powers and their opposite are {S, ⊥}-definable over Z. Applications to definability over Z and N are stated as corollaries of the main theorem.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 54 , Issue 4 , December 1989 , pp. 1253 - 1287
Copyright
Copyright © Association for Symbolic Logic 1989

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References

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