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On certain axiomatizations of arithmetic of natural and integer numbers. (English) Zbl 1432.03124

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 \(P\) 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.

MSC:

03F30 First-order arithmetic and fragments
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