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The arithmetics as theories of two orders. (English) Zbl 0555.03026

Orders: description and roles, Proc. Conf. Ordered sets appl., l’Arbresle/France 1982, Ann. Discrete Math. 23, 287-311 (1984).
[For the entire collection see Zbl 0539.00003.]
The paper is devoted to the study of properties of the natural order and the order of divisibility in arithmetic. After a survey of some classical results on nonstandard models of Peano arithmetic the author discusses the problem of incompleteness of arithmetic. The classical results of Gödel and Paris-Kirby are recalled and the undecidability of \(Th(N,S,\perp)\) (where \(S\) is the successor function and \(\perp\) denotes the relation of having no common prime divisors) and of related theories is proved. Next the saturation of models of arithmetic is considered. The main chapter 3 of the paper is devoted to the definability based on the natural order and on the order of divisibility \(|\) and their ”parts” \(S\) and \(\perp\). It contains some old and some new results. Among others two questions of J. Robinson concerning the definability of \(\{+,\cdot,<\}\) in terms of \(\{S,|\}\) and \(\{+,\perp\}\) (over integers including negative ones) are answered positively.
Reviewer: R.Murawski

MSC:

03F30 First-order arithmetic and fragments
03H15 Nonstandard models of arithmetic
03C62 Models of arithmetic and set theory

Citations:

Zbl 0539.00003