| Summary: | 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 <i>P</i> 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 <i>W</i> 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 <i>W</i> 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 <i>P</i> and <i>W</i> are equivalent and also that the systems of arithmetic based on <i>W</i> or on <i>P,</i> are categorical and consistent. There follows a set of intuitive axioms <i>PI</i> of integers arithmetic, modelled on <i>P</i> and proposed by B. Iwanuś, as well as a set of axioms <i>WI</i> of this arithmetic, modelled on the <i>W</i> axioms, <i>PI</i> and <i>WI</i> 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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