# On Certain Axiomatizations of Arithmetic of Natural and Integer Numbers

*Axioms*2019 (Deductive Systems) (2019)

**Abstract**

The systems of arithmetic discussed in this work are non-elementary theories. In this
paper, natural numbers are characterized axiomatically in two dierent 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.

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Archival date: 2019-09-18

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