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Open AccessArticle

On Certain Axiomatizations of Arithmetic of Natural and Integer Numbers

by Urszula Wybraniec-Skardowska
Department of Philosophy, Cardinal Stefan Wyszyński University in Warsaw, Wóycickiego 1/3, 01-938 Warsaw, Poland
Axioms 2019, 8(3), 103; https://doi.org/10.3390/axioms8030103
Received: 1 July 2019 / Revised: 14 August 2019 / Accepted: 1 September 2019 / Published: 4 September 2019
(This article belongs to the Special Issue Deductive Systems)
The systems of arithmetic discussed in this work are non-elementary theories. In this paper, natural numbers are characterized axiomatically in two different ways. We begin by recalling the classical set P of axioms of Peano’s arithmetic of natural numbers proposed in 1889 (including such primitive notions as: set of natural numbers, zero, successor of natural number) and compare it with the set W of axioms of this arithmetic (including the primitive notions like: set of natural numbers and relation of inequality) proposed by Witold Wilkosz, a Polish logician, philosopher and mathematician, in 1932. The axioms W are those of ordered sets without largest element, in which every non-empty set has a least element, and every set bounded from above has a greatest element. We show that P and W are equivalent and also that the systems of arithmetic based on W or on P, are categorical and consistent. There follows a set of intuitive axioms PI of integers arithmetic, modelled on P and proposed by B. Iwanuś, as well as a set of axioms WI of this arithmetic, modelled on the W axioms, PI and WI being also equivalent, categorical and consistent. We also discuss the problem of independence of sets of axioms, which were dealt with earlier. View Full-Text
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Wybraniec-Skardowska, U. On Certain Axiomatizations of Arithmetic of Natural and Integer Numbers. Axioms 2019, 8, 103.

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