Next Article in Journal / Special Issue
Logic of Typical and Atypical Instances of a Concept—A Mathematical Model
Previous Article in Journal
Generalized-Hypergeometric Solutions of the General Fuchsian Linear ODE Having Five Regular Singularities
Previous Article in Special Issue
Deductive Systems with Multiple-Conclusion Rules and the Disjunction Property
Open AccessArticle

On Certain Axiomatizations of Arithmetic of Natural and Integer Numbers

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
Keywords: axiomatizations of arithmetic of natural and integers numbers; second-order theories; Peano’s axioms; Wilkosz’s axioms; axioms of integer arithmetic modeled on Peano and Wilkosz axioms; equivalent axiomatizations; metalogic; categoricity; independence; consistency axiomatizations of arithmetic of natural and integers numbers; second-order theories; Peano’s axioms; Wilkosz’s axioms; axioms of integer arithmetic modeled on Peano and Wilkosz axioms; equivalent axiomatizations; metalogic; categoricity; independence; consistency
MDPI and ACS Style

Wybraniec-Skardowska, U. On Certain Axiomatizations of Arithmetic of Natural and Integer Numbers. Axioms 2019, 8, 103.

Show more citation formats Show less citations formats
Note that from the first issue of 2016, MDPI journals use article numbers instead of page numbers. See further details here.

Article Access Map by Country/Region

1
Back to TopTop