Abstract
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.
Highlights
The notion of natural numbers counts amongst the oldest, being one of the most universal abstract notions
Natural numbers belong to the fundamental subject of study of theoretical arithmetic concerned with defining all kinds of numbers, as well as studying their properties and relations between numbers of the same or different kinds
Presented theories will be non-elementary second-order theories and alphabet of languages which will assume two sorts of variables: individual variables and variables ranging over sets of individuals, i.e., natural numbers or integers, respectively
Summary
The notion of natural numbers counts amongst the oldest, being one of the most universal abstract notions. NA and arithmetic of integer numbers IA (the presentation is based on results originally published in Polish by various authors, and which, as a consequence of their being available only in Polish, are not known among the vast majority of mathematical logicians). Presented theories will be non-elementary second-order theories and alphabet of languages which will assume two sorts of variables: individual variables and variables ranging over sets of individuals, i.e., natural numbers or integers, respectively. We will start with the original axiomatization of NA proposed by Giuseppe Peano [1] by the set P of axioms on which is based the deductive system PA (the axiomatic non-elementary deductive theory; for short: the system PA) and will compare it with the little known axiomatization of the arithmetic NA by the set W of axioms, which was provided by Witold Wilkosz [2], a Polish logician, mathematician and philosopher of Kraków. We will give several metalogical theorems of the systems of arithmetic, which are presented
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