axioms Article On Certain Axiomatizations of Arithmetic of Natural and Integer Numbers Urszula Wybraniec-Skardowska Department of Philosophy, Cardinal Stefan Wyszyński University in Warsaw, Wóycickiego 1/3, 01-938 Warsaw, Poland; skardowska@gmail.com; Tel.: +48-22-569-6801 Received: 1 July 2019; Accepted: 1 September 2019; Published: 4 September 2019   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. 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 1. Introduction 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. Theoretical arithmetic deals with examination of different types of numbers and their axiomatization, including that of natural numbers and integers. While defining its notions, we base ourselves on second-order logic and set theory. In the paper, we will discuss some different axiomatizations of arithmetic of natural numbers 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. The deductive system based on Wilkosz's set W of axioms will be denoted by WA. Axioms 2019, 8, 103; doi:10.3390/axioms8030103 www.mdpi.com/journal/axioms Axioms 2019, 8, 103 2 of 14 Then, we will expand both sets P and W of axioms to the sets of axioms of arithmetic of integer numbers IA, which are modeled on them: the set of axioms PI by Iwanuś [3] and mine WI [4,5], which will be compared with each other and also with the set SI of axioms given by Sierpiński [6]. We will also give several metalogical theorems of the systems of arithmetic, which are presented. 2. Two Simple Axiomatizations of NA 2.1. Peano's Axioms for PA Historically, the first axiomatic system of arithmetic of natural numbers, which is characterized by unique simplicity, was that presented by Italian mathematician Giuseppe Peano in 1889, in his book in Latin [1]. The essential ideas of axiomatization of NA were first published by Dedekind [7] in 1888. Peano's axioms specify the ideas, but there can be no doubt concerning the originality of Peano's work (see [8], p. 101). Peano's original formulation of the axioms assumes the following three primitive notions: number (positive integer) N, unity 1 and the successor of a number; the modern set memberships relation ∈ comes from Peano's relation ε (is) that he used in [1] (see [8], chapter VII). The most modern formulations of Peano's axioms use 0 as the "first" natural number instead of 1 and the set of all natural numbers as N. In this work the successor of a number is a unary function defined on natural numbers and denoted by the symbol *. In modern presentations, Peano's axioms are written using the symbolism of mathematical logic and set theory. They are axioms of a non-elementary theory of natural numbers, including set theory. Recollecting them, we use the convention that the individual variables m, n, k, l, . . . ranging over the set N, while those of X, Y, Z . . . over subsets of the set N. Peano's axioms of the system PA are the following: P1. 0 ∈ N, P2. n* ∈ N, P3. n* , 0, P4. m* = n*⇒m = n, P5. 0 ∈ X ∧ ∀k ∈ X (k* ∈ X)⇒ N ⊆ X (the induction principle). Axioms P1–P4 are elementary ones, whereas axiom P5, called induction axiom, is an axiom of the 2nd order, a non-elementary one. In the first-order Peano arithmetic (elementary arithmetic) which is weaker than PA (see, e.g., [9], chapter II, section 7, chapter III, section 5 and [10], chapter 5), it is reformulated by the induction axiom schema. The induction axiom is applied in inductive proofs of theorems of the form T(n), where n denotes a natural number. If T(n) with the free variable n is an expression of arithmetic NA and T(0) is its true expression and from the assumption that T(k) is true for k ≥ 0 it follows that T(k*)-its truthfulness for number k*-then T(n) is true for any natural number of set N. Such proofs are based on the following schema of the rule of inductive proof of theorem T(n) for all n: T (0) T(k)⇒ T (k*) for any k ≥ 0 -----------– T(n) for any n ∈ N In proofs of theorems based on the set of axioms P, the following generalized theorems on induction are also made use of: T1. m ∈ X ∧ ∀k ∈ X (m ≤ k⇒ k* ∈ X)⇒ ∀n ∈ N (m ≤ n⇒ n ∈ X), T2. ∀m (∀k < m (k ∈ X)⇒m ∈ X))⇒ ∀n ∈ N (n ∈ X). In compliance with T1, if any set of natural numbers to which m belongs satisfies the condition that for each number k of the set, which is not smaller than m, its successor k* also belongs to this set, then to this set belong all the natural numbers not smaller than m. With m = 0, T1 = P5. With theorem T1 corresponds the rule of inductive proof based on the following schema: Axioms 2019, 8, 103 3 of 14 T(m) for m ∈N T(k)⇒T(k*) for k ≥m ----------– T(n) for all n ≥m, n ∈ N In compliance with T2, if each natural number m satisfies the condition: if any number smaller than m belongs to a given set of natural numbers, then m also belongs to this set, then each natural number does. With theorem T2 corresponds the rule of inductive proof based on the following schema: for each m ∈N T(k)⇒ T(m) for any k < m -----------– T(n) for all n ∈ N The proof of theorem T1 is based on the induction axiom P5 and the elementary theorems of system PA, whereas in the proof of theorem T2 the minimum principle and the elementary theorems of system PA are made use of. The former follows from the induction principle P5 and requires introducing additional definitions into system PA, including the definition of relation of less than, <, and that of non-greater than, ≤. It can be proved that the induction principle P5, the maximum principle Pmax and the minimum principle Pmin are equivalent to one another on the basis of the elementary theorems of system PA, since the following relations of implication hold: P5→ Pmax→ Pmin→ T2→ P5 where Pmax. In any non-empty set of natural numbers for which there is an upper bound element, there is the greatest number. Symbolically: ∃k (k ∈ X) ∧ ∃n∀m∈X (m ≤ n)⇒ ∃n∈X∀m∈X (m ≤ n). Pmin. In any non-empty set of natural numbers, there is the least number. Symbolically: ∃k (k ∈ X)⇒ ∃n∈X∀m∈X (n ≤m). Thus, we obtain the first metalogical theorem: MT1. The principles P5, Pmax and Pmin are mutually equivalent on the basis of the elementary theorems of system PA. Remark. Each of these non-elementary expressions could then be the only non-elementary axiom of arithmetic of natural numbers NA if-from it and suitably selected elementary axioms-each elementary theorem follows (cf. Słupecki et al. [11] and Sierpiński [6]). The principles Pmax and Pmin are noted in system PA by means of relations of less than, <, or non-greater than, ≤, but the former is defined by means of the operation of addition +. The definitions of the relation < and that of ≤ in system PA are the following: D3. m < n⇔ ∃k ∈ N \ {0} (m + k = n), D4. m ≤ n⇔m = n ∨m < n. The definitions of the operations of addition + and multiplication * are recursive in PA: D1a. m + 0 = 0, b. m + n* = (m + n)*. Axioms 2019, 8, 103 4 of 14 D2a. m*0 = 0, b. m*n* = m*n + m. These operations satisfy the well-known properties of a commutative semi-ring with unity (1 = 0*), and it can be proved that the relation < (relation ≤ ) in PA well-orders set N (we differentiate two well-known notions of a relation ordering a set: strict ordering (<) a set and weak ordering (≤) a set). The structure < N, +, ., 0, 1, ≤ > is an ordered commutative semi-ring. 2.2. Wilkosz's Axioms for System WA The primitive notions in Wilkosz's axiomatic system WA are: the set of natural numbers N and the relation of less than, <. We write Wilkosz's axioms, accepting that the variables m, n, l, k, . . . run over set N, X is a subset of N. W1. ∃n (n∈ N)-there is a natural number, W2. m , n⇒m < n ∨ n < m-trichotomy, W3. (m < n⇒ ~ (n < m))-anti-symmetry of relation <, W4. m < n ∧ n < k⇒m < k-transitivity of relation <, W5. m < n⇒m, n ∈ N-the field of relation < is set N, W6. ∃k (k ∈ X)⇒ ∃n∈X∀m∈X (n ≤m)-the minimum principle, W7. ∃k (k ∈ X) ∧ ∃n∀m∈X (m ≤ n)⇒ ∃n∈X∀m∈X (m ≤ n)-the maximum principle, W8. ~∃m∀n (m , n⇒ n < m)-there is not the greatest number in set N. It is easy to see that in system WA, the relation < well-orders set N (in the sense of strict order). Relation <-a primitive notion in Wilkosz's system WA-is a notion defined in Peano's system PA (see D3), and the primitive notions of system PA, which are not primitive ones in Wilkosz's system WA, are defined in it in the following way: (1) k = 0⇔ ∀n (k ≤ n)-0 is the least natural number, (2) k = n*⇔ k ∈{m ∈ N | n < m} ∧ ∀i ∈{m ∈ N | n < m}(k ≤ i)-n* is the least natural number among numbers greater than n. Relation ≤ less than or equal (not greater) has the following definition: m ≤ n⇔m = n ∨m < n. It can be proved that the definitions (1) and (2) are correct: there is precisely one natural number k satisfying the definiens of definition (1) and there is precisely one number k (the successor of number n), which satisfies the definiens of definition (2). In the proofs the axioms W2–W4 are used. Relying on, in Wilkosz's system WA, the definitions of zero and the successor function of a natural number, we can define the operations of addition + and multiplication, in the same way as in system PA (by means of definitions D1a,b and D2a,b). 2.3. Equivalence of the Deductive Systems PA and WA It needs reminding that, in accordance with Tarski's inferential definition of two equivalent sets of sentences of a deductive system (see [12]), two sets of sentences are equivalent if sets of all their consequences (deduced from them sentences) are equal. Thus, most often, for the equivalence of axiomatic deductive system the following definition (cf. [13,14]) is used: (*) Two axiomatic deductive systems are equivalent if the set of axioms and definitions of one of them is equivalent to the set of axioms and definitions of the other system, i.e., if each axiom and definition of one of them is a theorem or definition of the other system and the other way round-each axiom and definition of the other system is a theorem or definition of the first system. Let us note that Axioms 2019, 8, 103 5 of 14 (i1) Axioms of system WA are theorems of system PA since W1 follows directly from P1; W8 follows from the fact that in system PA there holds the theorem that n < n* for any n ∈ N, and n* ∈ N (P2); W3–W5 in PA follow from the theorem that N is a set ordered by the relation <; W6 and W7 (principles minimum and maximum, respectively) follow from the induction axiom P5 (see MT1). (i2) Definitions (1) and (2) of zero and the successor of a natural number in WA are theorems in system PA. (i3) Definitions of the operations addition and multiplication in system WA are the same as in system PA. (i4) Each axiom and definition of system WA is a theorem or definition in system PA. On the other hand (j1) Axioms of system PA are theorems of system WA, since from the correctness of definitions (1) and (2) in WA, in particular axioms P1 and P2 follow; also P3 is a theorem in WA, because if it would be possible that n* = 0, then it would follow from (1) that n* ≤ n and from (2) that n < n*, and hence that 0 ≤ n and n < 0, that is on the basis of W3: ~0 < n and n = 0, that is 0 < 0, which leads to contradiction according to W3; next, P4 follows from (2) and from the property of relation <, as one ordering set N. Axiom P5-the induction principle follows from those of maximum and minimum (W7 and W6; see MT1). (j2) Definition D3 of relation < in PA can be derived from axioms and definitions of system WA. (j3) Each axiom and definition of system PA is a theorem of system WA. From (i4) and (j3), in compliance with (*), we obtain the following metatheorem: MT2. The systems of PA and WA are equivalent. This equivalence was sketched in the booklet by Wilkosz [2] under the title Arytmetyka liczb całkowitych (The Arithmetic of Integers). Equivalence of Wilkosz's and Peano's systems was the subject of my MA thesis. 2.4. Independence of Axioms in Systems PA and WA As is well known from Gödel's first incompleteness theorem given in 1931 [15], no finite set of axioms of natural numbers is complete, or even each infinite, countable set of axioms of arithmetic is incomplete. There arises the problem, however, whether it is possible to reduce the number of axioms of PA and WA without depleting the set of theorems which can be proved about natural numbers. It can be shown that MT3a. The set of axioms of PA arithmetic system is independent (Sierpiński [6]). b. The set of axioms of WA system is dependent and can be reduced to the set: {W1, W3, W5, W6, W7, W8}. Axiom W2 follows from axiom W6, while axiom W4 follows from axioms W3, W5 and W6. Hence, it follows that the axioms of the theory of well-ordered sets in regard to relation < can be based on axioms W3, W5 and W6, while Wilkosz's system WA can be based on the axioms: W1′. ∃n (n∈ N)-there is a natural number, W2.' ∀m∃n (m ≤ n)-in set N there is not the greatest number, W3.' ∀m∀n (m < n⇔ ~ (n < m))-asymmetry of relation < in N, W4.' ∃k (k ∈ X)⇒ ∃n∈X∀m∈X (n ≤m)-the minimum principle, W5.' ∃k (k ∈ X) ∧ ∃n∀m∈X (m ≤ n)⇒ ∃n∈X∀m∈X (m ≤ n)-the maximum principle. To prove independence of the axioms one can, as it is known, use the method of interpretation, which consists in finding such an interpretation of primitive terms of the given system that makes all the axioms, apart from one, e.g., Ai, true at the interpretation. If we find it, then the given axiom Ai is independent from the others. Axioms 2019, 8, 103 6 of 14 2.5. Categoricity of Arithmetic Systems PA and WA Let us recall the definition of the notion of categoricity (see, e.g., [16–18]): (**) A deductive system is categorical if and only if all its models are isomorphic. As we mentioned, PA and WA as second-order, non-elementary systems, as well as elementary Peanos arithmetic, are not complete, yet we can show that they are categorical (cf. [7,10,18,19], chapter 8). A model of Peano's arithmetic system PA is each triple <N, 0, S > assigned to the triple <N, 0, * > of primitive terms of system PA, where N is an infinite set, 0 ∈N, and S: N→ N, which satisfies Peano's axioms P1–P5. A model of Wilkosz's arithmetic system WA is each tuple <N, Axioms 2019, 8, x FOR PEER REVIEW 6 of 14 2.4. Independence of Axioms in Systems PA and WA As is well known from Gödel's first incompleteness theorem given in 1931 [15], no finite set of axioms of natural numbers is complete, or even each infinite, countable set of axioms of arithmetic is incomplete. There arises the problem, however, whether it is possible to reduce the number of axioms of PA and WA without depleting the set of theorems which can be proved about natural numbers. It can be shown that MT3a. The set of axioms of PA arithmetic system is independent (Sierpiński [6]). b. The set of axioms of WA system is dependent and can be reduced to the set: {W1, W3, W5, W6, W7, W8}. Axiom W2 follows from axiom W6, while axiom W4 follows from axioms W3, W5 and W6. Hence, it follows that the axioms of the theory of well-ordered sets in regard to relation < can be based on axioms W3, W5 and W6, while Wilkosz's system WA can be based on the axioms: W1′. ∃n (n∈ N)-there is a natural number, W2.' ∀m∃n (m ≤ n)-in set N there is not the greatest number, W3.' ∀m∀n (m < n ⇔ ~ (n < m))-asymmetry of relation < in N, W4.' ∃k (k ∈ X)  ∃n∈X∀m∈X (n ≤ m)-the minimum principle, W5.' ∃k (k ∈ X) ∧ ∃n∀m∈X (m ≤ n)  ∃n∈X∀m∈X (m ≤ n)-the maximum principle. To prove independence of the axioms one can, as it is known, use the method of interpretation, which consists in finding such an interpretation of primitive terms of the given system that makes all the axioms, apart from one, e.g., Ai, true at the interpretation. If we find it, then the given axiom Ai is independent from the others. 2.5. Categoricity of Arithmetic Systems PA and WA Let us recall the definition of the notion of categoricity (see, e.g., [16–18]): (**) A deductive system is categorical if and only if all its models are isomorphic. As we mentioned, PA and WA as second-order, n n-el mentary ystems as well as elementary Peanos arithmetic, are not omplete, yet we can show that the are categorica (cf. [7,10,18,19], chapter 8). A model of Peano's arithmetic system PA is eac triple <N, 0, S > ass gned to the triple <N, 0, * > of primitiv terms of system PA, where N is an infin te set, 0 ∈N, and S: N → N, which satisfies Peano's axioms P1–P5. A model of Wilkosz's arithmetic system WA is each tuple <N, Հ > assigned to the tuple < N, < > of primitive terms f WA system, where N is an infinite, countable set and Հ a binary relation with the field N, which satisfies Wilkosz's axioms W1–W8 (W1′–W5′). (m1) Two models of PA: P1 = <N1, 01, S1 > and P2 = <N2, 02, S2 > are isomorphic if and only if there is bijection f: N1 → N2 such that f is homomorphism from P1 to P2, that is f(01) = 02 and f(S1(m)) = S2(f(m)) for any m ∈ N1. (m2) Two models of WA: W1 = <N1, Հ1 > and W2 = <N2, Հ 2 > are isomorphic if and only if there is bijection f: N1 → N2 being homomorphism from W1 to W2, that is m Հ1 n  f (m) Հ 2 f(n) for any m, n ∈ N1. Dedekind already in [7] proved that (m3) Each two models of arithmetic system PA are isomorphic. In the book by Słupecki et al. [11], there is a proof that (m2) Each two models of arithmetic system WA are isomorphic. > assigned to the tuple < N, < > of primitive terms of WA system, where N is an infinite, countable set and Axioms 2019, 8, x FOR PEER REVIEW 6 of 14 2.4. Independence of Axioms in Systems PA and WA As is well known from Gödel's first in ompleteness theor m given n 1931 [15], no f n te set of axioms of n tural numbers is complet , or even each infinite, countabl set of axioms of arithmetic is i complete. There arises the problem, ow ver, whether it is ossible to reduce the of axioms of PA and WA wi hout depleting the set of theorems which can be proved about natural numbers. It can be shown that MT3a. The set of axioms of PA arithmetic sys em is indepen ent (Sierpiń ki [6]). b. The set of axioms of WA system is dependent and can be reduced to the set: { 1, W3, W5, W6, W7, W8}. Axi m W2 follows from axiom W6, while axiom W4 follows from axioms W3, W5 and W6. Hence, it follows that the axioms of the theory of well-ordered sets in r gard to relation < can be based on axioms W3, W5 and W6, while Wilkosz's system WA can be based on the axioms: W1′. ∃n (n∈ N)-there is a natural number, W2.' ∀m∃n (m ≤ n)-in set N there is not the greatest number, W3.' ∀m∀n (m < n ⇔ ~ (n < m))-asymmetry of relation < in N, W4.' ∃k (k ∈ X)  ∃n∈X∀m∈X (n ≤ m)-the minimum principle, W5.' ∃k (k ∈ X) ∧ ∃n∀m∈X (m ≤ n)  ∃n∈X∀m∈X (m ≤ n)-the maximum principle. To prove independe ce of th xi ms one can, as it is known, us the method of int rpretation, which consists in findin such an interpretation of primitive terms of giv n system that makes all the axioms, apart from one, e.g., Ai, true at the interpretation. If we find it, then the given axiom Ai is independent from the others. 2.5. Categoricity of Arithmetic Systems PA and WA Let us recall the definition of the otion of c tegoricity ( ee, e.g., [16–18]): (**) A dedu tive syst m s categorical if and o ly if all its models are i omorphic. As we mentioned, PA and WA as second-or er, nonementary syst ms, as well as elementary Peanos arithmetic, are not complete, yet we can sh w that they are ca egorical (cf. [7,10,18,19], chapter 8). A model of Pean 's arit etic t PA is each triple <N, 0, S > assigned to the triple <N, 0, * > of primitive terms of system PA, where N is an infinite set, 0 ∈N, and S: N → N, which satisf es Peano's axioms P1–P5. A model of Wilkosz' arithmetic y tem WA is each tuple <N, Հ > assigned to he tuple < N, < > of prim tive t rms of WA ystem, where N is an infinite, co t l Հ a binary relation with the field N, which satisfies Wilkosz's axioms W1–W8 (W1′–W5′). (m1) Two models of PA: P1 = <N1, 01, S1 > and P2 = <N2, 02, S2 > are isomorphic if and only if there is bijection f: N1 → N2 such that f is homomorphism from P1 to P2, that is f(01) = 02 and f(S1(m)) = S2(f(m)) for any m ∈ N1. (m2) Two mod ls of WA: W1 = <N1, Հ1 > and W2 = <N2, Հ 2 > are isomorphic if and only if there is bijection f: N1 → N2 being homomorphism from W1 to W2, that is m Հ1 n  f (m) Հ 2 f(n) for any m, n ∈ N1. Dedekind already in [7] proved that (m3) Each two models of arithmetic system PA are isomorphic. In the book by Słupecki et al. [11], there is a proof that (m2) Each two models of arithmetic system WA are isomorphic. relation with the field N, which satisfies Wilkosz's axioms W1–W8 (W1′–W5′). (m1) Two models of PA: P1 = <N1, 01, S1 > and P2 = <N2, 02, S2 > are isomorphic if and only if there is bijection f: N1 → N2 such that f is homomorphism from P1 to P2, that is f(01) = 02 and f(S1(m)) = S2(f(m)) for any m ∈ N1. (m2) ls f : 1 1, Axioms 2019, 8, x FOR PEER REVIEW 6 of 14 2.4. Independence of Axioms in Systems PA and WA As is well known from Gödel's first incompleteness theorem given in 1931 [15], no finite set of axioms of natural numbers is complete, or even each infinite, countable set of axioms of arithmetic is incomplete. There arises the problem, however, hether it is possible to reduce the number of axioms of PA and WA without depleting the set of theorems which can be proved about natural numbers. It can be shown that MT3a. The set of axioms of PA arithmetic system is ndependent (Sierpiński [6]). b. The set of axioms of WA system is dependent and can be reduced to the set: {W1, W3, W5, W6, W7, 8} Axiom W2 follows from axiom 6, while axiom W4 follows from axioms W3, W5 and W6. Hence, it follows that the axioms of the theory of well-ordered sets in regard to relation < can be based on axioms W3, W5 and W6, while Wilkosz's system WA can be based on the axioms: W1′. ∃n (n∈ N)-there is a natural number, W2.' ∀m∃n (m ≤ n)-in set N there is not the greatest number, W3.' ∀m∀n (m < n ⇔ ~ (n < m))-asymmetry of relation < in N, W4.' ∃k (k ∈ X)  ∃n∈X∀m∈X (n ≤ m)-the minimum principle, W5.' ∃k (k ∈ X) ∧ ∃n∀m∈X (m ≤ n)  ∃n∈X∀m∈X ( ≤ n)- he maximum principle. To prove independence of the axioms on can, as it is known, use the metho of interpretation, which consists in finding such an interpretation of primitive terms of the given system that makes all the axioms, apart from one, e.g., Ai, true at the interpretation. If we find it, then the given axiom Ai is independent from the others. 2.5. Categoricity of Arithmetic Systems PA and WA Let us recall the definition of the notion of categoricity (see, e.g., [16–18]): (**) A deductive system is categorical if and only if all its odels are isomorphic. As we mentioned, PA and WA as second-or er, non-ele entary systems, as well as elementary Peanos arithmetic, are not complete, yet we can show that they are categorical (cf. [7,10,18,19], chapter 8). A model of Peano's arithmetic system PA is each triple <N, 0, S > assigned to the triple <N, 0, * > of primitive terms of system PA, here is an infinite set, 0 ∈N, and S: N → N, which satisfies Peano's axioms P1–P5. A model of Wilkosz's arithmetic syste WA is each tuple < , Հ > assigned to the tuple < N, < > of primitive terms of WA system, where N is an infinite, countable set and Հ a binary relation with the field N, which satisfies Wilkosz's axioms W1–W8 (W1′–W5′). (m1) Two models of PA: P1 = <N1, 01, S1 > and P2 = <N2, 02, S2 > are isomorphic if and only if there is bijection f: N1 → N2 such that f is homomorphism from P1 to P2, that is f(01) = 02 and f(S1(m)) = S2(f(m)) for any m ∈ N1. (m2) Two models of WA: W1 = <N1, Հ1 > and W2 = <N2, Հ 2 > are isomorphic if and only if there is bijection f: N1 → N2 being homomorphism from W1 to W2, that is m Հ1 n  f (m) Հ 2 f(n) for any m, n ∈ N1. Dedekind already in [7] proved that (m3) Each two models of arithmetic system PA are isomorphic. In the book by Słupecki et al. [11], there is a proof that (m2) Each two models of arithmetic system WA are isomorphic. 1 > and 2 = <N2, Axioms 2019, 8, x FOR PEER REVIEW 6 of 14 2.4. Independence of Axioms in Systems PA and WA As is well know from Göd l's first incompleteness theorem given in 1931 [15], no finite set of axioms of natural numb rs is comple or even each in inite, countable set of axioms of arithmetic is incomplete. There arises the problem, however, wh ther it is possi le t reduce the number of axioms of PA and WA with ut depleting the set of theorems which c n be proved about natural numbers. It can be shown that MT3a. The set of axioms of PA arithmetic system s independent (Sierpiński [6]). b. The set of axioms of WA system is dependen and can be reduced to the set: {W1, W3, W5, W6, W7, W8}. Axiom W2 follows fro axiom W6, hile axiom W4 follows from axioms W3, W5 and W6. Hence, it follows that the axioms of the theory of well-ordered sets in regard to relation < can be based on axioms W3, W5 and W6, while Wilkosz's ystem WA can be based on the axioms: W1′. ∃n (n∈ N)-th e is a natural number, W2.' ∀m∃n (m ≤ n)-in set N there is not the greatest number, W3.' ∀ ∀n (m < n ⇔ ~ (n < m))-asymmetry of relation < in N, W4.' ∃k (k ∈ X)  ∃n∈X∀ ∈X (n ≤ m)-the minimum principle, W5.' ∃k (k ∈ X) ∧ ∃n∀m∈X (m ≤ n)  ∃n∈X∀m∈X ( ≤ n)-the maximum principle. To prove independence of the ax oms one can, as it is kn wn, use the method of interpretation, which onsists in finding such an interpretati n of pr mitive t r s of the given system that makes all the axioms, apa t from one, e.g., Ai, true at the terpretation. If w find it, then the given axiom Ai is independent from the others. 2.5. Categoricity of Arithmetic Systems PA and WA Let us recall the definition f the notion of categoricity (see, e.g., [16–18]): (**) A d ductive system is categorical if and only if all its models are isomorphic. As we mentioned, PA and WA as second-order, non-elementary systems, as well as elementary Peanos arithmetic, are not complete, yet we can sh that they are categorical (cf. [7,10,18,19], chapter 8). A mod l of Peano's arithmetic syst m PA is each triple <N, 0, S > assigned to the triple <N, 0, * > of pri itive terms f system PA, where N is an infinite set, 0 ∈ , and S: N → N, which satisfies Peano's axioms P1–P5. A model of Wilkosz's arithmetic system WA i each tuple <N, Հ > assigned to the tuple < N, < > of primi ive ter s of WA system, where N is an infinite, cou abl set and Հ a binary elation with the field N, which sati fies Wilkosz's axioms W1–W8 (W1′–W5′). (m1) Two models of PA: P1 = < 1, 01, S1 > and P2 = <N2, 02, S2 > are isomorph c if and only if there is bijection f: N1 → N2 suc that f is homomorphism from P1 to P2, that is f(01) = 02 and f(S1( )) = S2(f(m)) for any m ∈ N1. (m2) Two models of WA: W1 = <N1, Հ1 > and W2 = <N2, Հ 2 > are isomorphic if and only if there is bijection f: N1 → N2 being hom morp ism from W1 to W2, that is m Հ1  f (m) Հ 2 f(n) for any m, n ∈ N1. Dedekin already in [7] proved that (m3) Each two models of arithmetic system PA are isomorphic. In the book by Słupecki et al. [11], there is a proof that (m2) Each two models of arithmetic system WA are isomorphic. 2 > are isomorphic if and only if there is b jection f: N1 → N2 being homomorphism from W1 to W2, that is m Axioms 2019, 8, x FOR PEER REVIEW 6 of 14 2.4. Independence of Axioms in Systems PA and WA As is well known from Gödel's first inco pleteness th orem given in 1931 [15], no finite set of axioms of natural nu bers i complete, or even each infinite, countable set of axioms of arithmetic is incomplete. Th re aris s the problem, ho ever, whethe it is possible to reduce the number of axioms of PA and WA without depleting the set of theorems which can be proved about natural numbers. It can be shown t at MT3a. The set of axioms of PA a ithmetic syst m is independent (Sierpiński [6]). b. The set of axioms of WA sys em is dependent and can be reduced to the set: {W1, W3, W5, 6, W7, W8}. Axiom W2 foll ws from axio W6, hile axiom W4 follows from axioms W3, W5 and W6. Hence, it follows t at the axioms of the theory f well-or ered s ts in regard to relation < can be based on axioms W3, W5 and W6, while Wilkosz's system WA can be based on the axioms: W1′. ∃n (n∈ N)-there is a natural number, W2.' ∀m∃n (m ≤ n)-in set N there is not the greatest number, W3.' ∀m∀n (m < n ⇔ ~ (n < m))-asymmetry of relation < in N, W4.' ∃k (k ∈ X)  ∃n∈X∀m∈X (n ≤ m)-the minimum principle, W .' ∃k (k ∈ X) ∧ ∃n∀m∈X (m ≤ n)  ∃n∈X∀m∈X (m ≤ n)-the maximum principle. To pr ve independ nce of the axio s one can, as i is nown, us the method of interpretation, which consists in finding such an interpretation of primitive erms of the given system that makes all the axioms, apa t from one, e.g., Ai, true at the interpretation. If we find it, then the given axiom Ai is independent from the others. 2.5. Categoricity of Arithmetic Systems PA and WA L t u reca l the def niti n of th n tion of categoricity (see, e.g., [16–18]): (**) deductive sy tem is categorical if nd only if all its odels are isomorphic. As we mentioned, PA nd WA as second-order, non-elementary systems, as well as elementary Peanos arithmetic, are not complete, yet we can show that they are categorical (cf. [7,10,18,19], chapter 8). A model of Peano's arithm ic syste PA s each triple <N, 0, S > assigned to the triple <N, 0, * > of primitive terms of system PA, where N is an infinite set, 0 ∈N, and S: N → N, which satisf es Peano's axioms P –P5. A model of Wilkosz's arithmetic sys A is each tuple <N, Հ > assigned to the tuple < N, < > of pr mitiv terms of WA syst m, where N is an in inite, countable se and Հ a binary relation with the field N, which satisfies Wilkosz's axioms W1–W8 (W1′–W5′). (m1) Two models of A: P1 = <N1, 01, S1 > and P2 = <N2, 02, S2 > are isomorphic if and only if there is bijection f: N1 → N2 such that f is homomo phis f om P1 to P2, that is f(01) = 02 and f(S1(m)) = S2(f(m)) for any m ∈ N1. ( 2) Two models of WA: W1 = <N1, Հ1 > and W2 = <N2, Հ 2 > are isomorphic if and only if there is bijection f: N1 → N2 being homomorphism from W1 to W2, that is m Հ1 n  f (m) Հ 2 f(n) for any m, n ∈ N1. Dedekind already in [7] proved that (m3) Each two models of arithmetic system PA are isomorphic. In he b ok by Słupecki et al. [11], there is a roof that (m2) Each two models of arithmetic system WA are isomorphic. 1 n⇒ f (m) Axioms 2019, 8, x FOR PEER REVIEW 6 of 14 2.4. Independence of Axioms in Systems PA and WA As is well k own from Gödel's first ncompl teness theo em given in 1931 [15], no finite set of axioms of natural numb rs is com lete, o even each infinit , countable set of axioms of arithmetic is incomplete. There aris s the problem, however, wheth r i is possibl to reduce the number of axioms of PA and WA without d pleting the set of theorems which can be proved about natural numbers. It can be shown that MT3a. Th et of axioms of PA arithmetic system is inde endent (Sierpiński [6]). b. The s t of axioms of WA yst is depe d nt and can be reduced to the set: { 1, 3, 5, W6, W7, W8}. Axio W2 foll ws from xi m W6, while xiom W4 foll w from axioms 3, W5 and W6. H nce, it follows that th axioms th theory of well-ordered sets in regard to relation < can be ba ed on axioms 3, W5 and W6, while Wilkosz's syst m WA can b based on the axioms: W1′. ∃n (n∈ N)-there is a natural number, W2.' ∀m∃n (m ≤ n)-in set N there is not the greatest number, W3.' ∀m∀n (m < n ⇔ ~ (n < m))-asymmetry of relation < in N, W4.' k (k ∈ X)  ∃n∈X∀m∈X (n ≤ )-the minimum principle, W5.' ∃k (k X) ∧ ∃n∀m∈X (m ≤ n)  ∃n∈X∀m∈X (m ≤ n)-the maximum principle. To prov indep ndence of the xioms one can, as i is known, use the method of interpretation, which co sists in fi ding suc an interpretation of primi ive t rms of he given system that makes all the axi s, apart from one, e.g., Ai, rue a the interpretation. If we find it, then the given axiom Ai is independent from the others. 2.5. Categoricity of Arithmetic Systems PA and WA L t us recall the d f the n tion of categoricity (see, e.g., [16–18]): (**) A d duc ive is categoric l if and onl if all its models are isomorphic. As we mentio ed, PA and WA as secondrd , non-element ry systems, as well as elementary P anos arithmetic, ar not co plete, yet we can show that they are categorical (cf. [7,10,18,19], chapter 8). A m del of Peano' arithmet c yst PA is each triple <N, 0, S > assigned to the triple <N, 0, * > of primitive terms of syst m PA, where N is an infinite set, 0 ∈N, and S: N → N, which satisfies Peano's axioms P1–P5. A m del of ilkosz's a ithmetic system WA is each tupl <N, Հ > assign d o the tuple < N, < > of primitive ter s of WA syste , whe e N is an infinite, countable set and Հ a binary relation with t e field N, which satisfies W lkosz's axioms W1–W8 (W1′–W5′). (m1) Tw models of PA: P1 = <N1, 01 S1 > and P2 = <N2, 02, S2 > are isomorphic if and only if there is bijection f: N1 → N2 such that f is homomorphism from P1 to P2, that is 0 ) = 02 and f(S1(m)) = S2(f(m)) for any m ∈ N1. ( 2) Two models f WA: W1 = <N1, Հ1 > and W2 = <N2, Հ 2 > are isom rphic if and only if there is bijection f: N1 → N2 being homomorphism from W1 to W2, that is m Հ1 n  f (m) Հ 2 f( ) for any m, n ∈ N1. Dedekind al eady in [7] proved that (m3) Each two models of arithmetic sy te PA are isomorphic. In the book by Słupecki et al. [11], there is a proof that (m2) Each two models of arithmetic syste WA are isomorphic. 2 f(n) for any m, n ∈ N1. Ded kind lready in [7] proved that ( 3) Each two m dels f arithmetic system A are isomorphic. In the book by Słu e ki et al. [11], there is a proof that (m4) ach t o o els of arith etic syste W are iso orphic. Hence, w have the metal gic cor ll ry: MT4. The d ductive systems PA and WA of na ural nu bers arithmetic NA are categorical; they are in power א0, so they are aleph-null categorical systems. Thus, Peano's and Wilkosz's second-order systems have only one model, up to isomorphism. This is not so when we consider the systems of arithmetic of natural numbers as systems (elementary theories) of the first-order. According to he upw d Löwenheim–Skolem's theorem, there are non-standard models of Peano's elementary arithmetic system of all infinite cardinality (see e.g., [9], chapter III, section 5, [20], chapterVI). 2.6. Set-Theoretical Models for PA and WA Peano's arithmetic possesses a "natural" set-theoretical model deriving from Frege. Let N be an infinite set of all cardinal numbers of finite subsets of any (infinite) set U, i.e., N = {card(X) | X ∈ Fin(U)}, where Fin(U) is the smallest family of sets to which the empty set ∅ belongs and which is closed under the relation S: X S Y⇔ ∃x ∈ U \ X (Y = X ∪ {x)) for any X, Y ∈ Fin (U). The formal definition of the set Fin (U) is the following: Fin (U) = ∩ { A ⊆ P(U) ∣∣∣ ∅ ∈ A ∧ ∀X ∈ A ∃Y ∈ A (X S Y ⇒ Y ∈ A)}. (mP) The set-theoretical model for PA is the triple < N, card (∅), S* >, where for m = card(X) and X S Y, S*(m) = m + 1 = card(Y), for X, Y ∈ Fin (U). (mW) The set-theoretical model for WA is the triple < N, ≺ >, where ≺ is the relation of less than for the cardinal numbers of set N: m ≺ n⇔m  n ∧ n ,m, Axioms 2019, 8, 103 7 of 14 m  n⇔ ∃X,Y (card(X) = m ∧ card(Y) = n ∧ ∃Z ⊆ Y(card (Z) = m)). From (mP) and (mW) we get two next metalogic corollaries: MT5. PA and WA systems are consistent. (since it follows from the theorem of categoricity (MT4) that all theorems of these systems are true, because they are true in each model of these systems). MT6. Systems PA and WA are (treated as) fragments of set theory. As we know, the theorem MT6 is of great importance to studies on the foundations of mathematics. 3. Simple Axiomatizations of Arithmetic of Integers, Based on Systems PA and WA Axiomatic systems for integer arithmetic IA are most often based on notions of operations of addition and multiplication defined on the set I of integers. In this part of the work, we will give an axiomatization of integer arithmetic IA modelled on the systems PA and WA respectively for the arithmetic of natural numbers NA, extending these systems accordingly and comparing them with Sierpiński's system SIA [6], including addition and multiplication as its primitive notions. 3.1. Iwanuś's Axioms for IA, Modelled on the Axioms of System PA We will give here two systems of axioms proposed by Bolesław Iwanuś [3] for IA system. They are interesting due to their intuitive character. The first system based on them will be denoted as P1IA, and the other one-P2IA. The primitive notions of P1IA are: set N* of all non-negative integers, set *N of all non-positive numbers, integer 0 and two unary operations in N*∪*N of successor and predecessor of an integer. The successor and the predecessor of integer i will be denoted as i* and *i, respectively. In the intuitive meaning, i* = i + 1 and *i = i – 1. We assume that i, j, k, l, . . . are variables ranging over the set N*∪*N, while variables A, B, C . . . range over the subsets of this set. 3.1.1. Axioms of System P1IA Are the Symmetric Axioms for Numbers of the Sets N* and *N: A*1. 0 ∈ N*, *A1. 0 ∈*N, A*2. i ∈ N*⇒ i* ∈ N*, *A2. i ∈ *N⇒ *i ∈ *N, A*3. i ∈ N*⇒ i* , 0, *A3. i ∈ *N⇒ *i , 0, A*4. 0 ∈ A ∧ ∀i ∈ A (i* ∈ A)⇒ N* ⊆ A, *A4. 0 ∈ A ∧ ∀i ∈ A (*i ∈ A)⇒ *N ⊆ A. A5. i ∈ N*∪*N⇒ *(i*) = i = (*i)*. Axioms A*1–A*3 and *A1–*A3 are modelled on those of Peano (P1–P3). Axioms A*4 and *A4 correspond to that of induction P5. Axiom A5 establishes relations between the successor and the predecessor operation and does not allow identification of these notions with each other, nor identification of sets N* and *N. The set I of all integers is defined as follows: D0. I = N* ∪ *N. The counterparts of Peano's axiom P4 i, j ∈ N* ∧ i* = j*⇒ i = j and i, j ∈ *N ∧ *i = *j⇒ i = j are direct consequences of A5. It is easy to notice that with the assumption that the set I of integers is a primitive notion of the system of arithmetic IA, the symmetrical axioms of P1IA can be replaced by weaker ones, deriving from Słupecki in [11]: A1. 0 ∈ I, A2. i ∈ I⇒ i*,*i ∈ I, A3. i ∈ I⇒ i* , i, A4, A ⊆ I ∧ 0 ∈ A ∧∀i ∈ A (i*, *i ∈ A)⇒ I =A, A5. i ∈ I⇒ (*i)* = *(i*) = i, In this system, there are theorems that, to a certain extent, are similar to Peano's axiom A3, which are in force: Axioms 2019, 8, 103 8 of 14 (i ∈ I ∧ i , *0)⇒ i* , 0, (i ∈ I ∧ i , 0*)⇒ *i , 0. In Iwanuś's system P1IA, there are the following definitions of the operations of: addition +, subtraction − and multiplication: DI1 a. i + 0 = i, DI2 a. i – 0 = i, DI3a. i*0 = i, b. i + j* = (i + j)*, b. i – j* = *(i j), b. i*j* = i*j + i, c. i +*j = * (i + j), c. i –*j = (i j) *, c. i**j = i*j – i. It is assumed that 1 = 0* and it is proved that I1. *N = (I – N*) ∪ {0}, I2. k = j – i⇔ i + k = j, I3. i* = i + 1, I4. *i = i –1. In proofs of the theorems of system P1IA the following meta-theorem is made use of: MT7. If α is an expression of system P1IA, in which-beside primitive notions-there are exclusively the defined terms + and ., then α is a theorem of this system if expression αd, dual with respect to α, is a thesis of this system; expression αd is dual to α, when the terms: N*, *N, ( )*, *( ), +, which occur in it, are substituted in each place of their appearance with the following ones, respectively: *N, N*,*( ), ( )*, +,. In proofs of theorems on the basis of axioms A*4 and *A4, the following rules of mathematical induction for integers based on the given below schemata are applied: T(0) T(0) T(k)⇒T(k*) for any k ≥ 0 T(k)⇒ T(k*) ∧ T(*k) for any k ----------- -------------T(i) for any i ∈ N* T(i) for any i ∈ I Remark 1. On the basis of system P1IA one can prove all the axioms of the commutative ring. The inequality relation less-than, <, in I is determined by the following definition added to P1IA: DI4. i < j⇔ ∃k ∈ N*\{0} (i + k = j). Remark 2. In system P1IA, one can prove all the theorems of arithmetic of integers IA relating to relation <. 3.1.2. The Other System of Arithmetic of Integers Built by Iwanuś [3] and Modelled on System PA The system is denoted by P2IA and based only on the following three primitive notions: set I of all integers, the function of successor * and number 0. The following formulas are the axioms of system P2IA: (I1) i ∈ I⇒ ∃j ∈ I (i = j*), (I2) i, j ∈ I ∧ i* = j*⇒ i = j, (I3) ∃A ⊆ I (0 ∈ A ∧ ∀i ∈A (i*∈ A ∧ i* , 0), (I4) A ⊆ I ∧ 0 ∈ A ∧ ∀i ∈A (i* ∈ A ∧∃j ∈A (i = j*))⇒ I ⊆ A. Axiom I3 assumes the existence of a certain subset of set I, about which-on the base of the above accepted set of axioms-it can be proved that it is isomorphic due to function * to the set of all natural numbers. Axiom I4 is a postulate of induction in the set of integers. Axioms 2019, 8, 103 9 of 14 If we introduce into system P2IA still one more primitive term N (as a name of a subset of set I which is isomorphic to the set of natural numbers), then axiom I3 can be substituted with the following set of axioms: I3a. N ⊆ I, b. 0 ∈ N, c. i ∈ N⇒ i* ∈ N, d. i ∈ N⇒ i* , 0. Axiom I3 is weaker than axioms I3a–d, because I3 follows from these axioms, although not all of I3a–d follow from I3. In P2IA system the primitive notions of P1IA system are defined in the following way: DI1′. i, j ∈ I⇒ (*i = j⇔ i = j*), DI2′. i ∈ N*⇔ ∀A ⊆ I (0 ∈ A ∧∀j ∈A (j* ∈ A))⇒ i ∈ A, DI3′. i ∈ *N⇔ ∀A ⊆ I (0 ∈ A ∧∀j ∈A (*j ∈ A))⇒ i ∈ A. All the remaining definitions of system P1IA are the same in system P2IA. Iwanuś proves that MT8. Systems P1IA and P2IA are equivalent. B. Iwanuś also proves in [3] that these systems are equivalent to Sierpiński's system of arithmetic of integers SIA [6], based on primitive notions: the set I, operations of addition + and multiplication *, zero 0, one 1 and the set N*, satisfying the axioms of the ring without zero divisors: R1. i, j ∈ I⇒ i + j ∈ I ∧ i * j ∈ I, R2. i, j, k ∈ I⇒ i + j = j + i ∧ i * j = j * i ∧ (i + j) + k = i + (j + k) ∧ (i * j)*k = i*(j * k) ∧ i*(j + k) = i * j + i * k, R3. ∀i, j ∈I ∃ k∈ I(i + k = j), R4. ∀i ∈ I (i + 0 = i) ∧ ∀i ∈ I (i*1 = i) ∧ 1 ∈ I, R5. i, j ∈ I ∧ i * j = 0⇒ i = 0 ∨ j = 0, R6. N* ⊂ I, R7. 0 ∈ N*, R8. i ∈ N*⇒ i +1 ∈ N*, R9. 0 ∈ A ∧ ∀i ∈ A (i* ∈ A)⇒ N* ⊆ A, R10. ∀i ∈ A\N* ∃j ∈ N* (i + j = 0). Definitions of primitive terms of system P1IA are introduced into system SIA as follows: DS1. *N = (I\N*) ∪ {0}, DS2. k = j – i⇔ i + k =j, DS3. i* = i + 1, DS4. *i = i – 1. Definition DI4 of relation < of system P1IA is the same as in system SIA. MT9. System P1IA (P2IA system), modelled on Peano's system of natural numbers arithmetic PA, and system SIA are equivalent. 3.2. Axioms of the System of Integer Arithmetic WIA Modelled on Wilkosz's System WA The primitive notions of the system of integer arithmetic WIA, modelled on Wilkosz's system WA, are the following: set I of all integers, integer zero 0 and less-than relation < in set I. The relation of weak inequality ≤ is determined by the definition (i, j, k, . . . run over I): D0. i ≤ j⇔ i < j ∨ i = j. The axioms of system WIA which are presented by Wybraniec-Skardowska [4,5] are the following expressions: W′1. 0 ∈ I, W′2. i, j ∈ I⇒ (i < j ∨ i = j ∨ j < i), Axioms 2019, 8, 103 10 of 14 W′3. i, j ∈ I⇒ (i < j⇒ ~ (j < i)), W′4. i, j, k ∈ I ∧ (i < j ∧ j < k)⇒ i < k, W′5. ∀i ∈ I ∃j ∈ I (i < j) – in I there is not the greatest number, W′6. ∀i ∈ I ∃j ∈ I (j < i) – in I there is not the smallest number, W′7. A ⊆ I ∧ ∃i ∈ A ∃i ∈ I ∀j ∈ A (i < j)⇒ ∃i ∈ A∀j ∈ A (i ≤ j), W′8. A ⊆ I ∧ ∃i ∈ A ∃i ∈ I ∀j ∈ A (j < i)⇒ ∃i ∈ A∀j ∈ A (j ≤ i). According to W'7, in each non-empty set of integers, which has a lower bound, there is the smallest number, while, according to W'8, in each non-empty set of integers, which has an upper bound, there exists the greatest number. The content of axioms W'7 and W'8 is close to the principles of minimum and maximum of arithmetic WA. The axioms of system WIA state that relation < orders set I, yet do not state that it well-orders the set. In system WIA one can define the notion of successor and that of predecessor of an integer as well as the notions of sets N* and *N, which are primitive notions in Iwanuś's system P1IA. Let us note first that in system WIA it is possible to prove the theorem: A ⊆ I ∧ ∃i ∈ A ∧ ∃i ∈ I∀j ∈ A (i < j)⇒ ∃1k ∈ A∀j ∈ A (k ≤ j). (1) Condition (1) allows introducing correctly the definition of minimum in set A: DW1. A ⊆ I ∧ ∃i ∈ A ∧ ∃i ∈ I∀j ∈ A (i ≤ j)⇒ (k = min (A)⇔ k ∈ A ∧ ∀j ∈ A (k ≤ j)). It follows from Condition (1) and DW1 that there is a unique minimum, min (A), when A ⊆ I, A , ∅ and set A has a lower bound. Let G(i) = {j∈ I | i < j}. Hence G(i) , ∅ (see W'5) and ∃1k ∈G(i)∀j ∈G(i) (k ≤ j) (see Condition (1)); then on the basis of DW1 ∃1k ∈ I (k = min (G(i)). (2) The successor of an integer i is introduced by means of the definition: DW2. i* = min(G(i)) – i* is the smallest integer which is greater than i. DW3. N* = {i ∈ I | 0 ≤ i}. The following corollary which is dual to Condition (1): A ⊆ I ∧ ∃i ∈ A ∧ ∃i ∈ I∀j ∈ A (j < i)⇒ ∃1k ∈ A∀j ∈ A (j ≤ k). (3) permits introducing the definition of maximum of a certain set of integers: DW4. A ⊆ I ∧ ∃i ∈ A ∧ ∃i ∈ I∀j ∈ A (j ≤ i)⇒ (k = max(A)⇔ k ∈ A ∧∀j ∈ A (j ≤ k)). Let L(i) = {j∈ I | j < i}. Hence L(i) , ∅ (see W'6) and ∃1k ∈ L(i)∀j ∈ L(i) (j ≤ k) (see Condition (3)) then on the basis of DW4 the predecessor of integer i is defined as the greatest integer less than i, that is DW5. *i = max(L(i)), and set *N is defined as follows DW6. *N = {i ∈ I | i ≤ 0}. 3.3. Equivalence of Systems P1IA and WIA Remark 3. With the definitions of the primitive notions of system P1IA, given in system WIA, all the axioms and definitions P1IA become theorems of definitions in system WIA. Axioms 2019, 8, 103 11 of 14 The definitions of addition and multiplication, which are accepted in P1IA are the same in system WIA, while definition DI4 of relation < accepted in system P1IA is a theorem in system WIA. Thus, it follows from Remarks 3 and 2 and MT8 that MT10. System WIA is equivalent to those of Iwanuś P1IA and P2IA. It follows from the above and MT9 that MT11. All the systems of integer arithmetic: WIA, P1IA, P2IA and SIA are mutually equivalent. In particular, MT12. System P1IA modelled on Peano's system PA and system WIA modelled on Wilkosz's system WA are equivalent. 3.4. Independence of the Axioms in P1IA and WIA The axioms of the integer arithmetic system P1IA can, as Iwanuś proved, be reduced by one axiom A*3 or *A3. If we found an axiom system of P1IA on those of A*1–A*4 and *A1, *A2 and *A4, then axiom *A3 can be proved. It follows from A*1, A*2, and A5 as well as from theorems of this system: *0 Axioms 2019, 8, x FOR PEER REVIEW 11 of 14 Condition (1) allows introducing correctly the definition of minimum in set A: DW1. A  I  i  A  i  Ij  A (i  j)  (k = min (A)  k  A  j  A (k  j)). It follows from Condition (1) and DW1 that there is a unique minimum, min (A), when A  I, A   and set A has a lower bound. Let G(i) = {j I | i < j}. Hence G(i)   (see W'5) and 1k G(i)j G(i) (k  j) (see Condition (1)); then on the basis of DW1 1k  I (k = min (G(i)). (2) The successor of an integer i is introduced by means of the definition: DW2. i* = min(G(i)) – i* is the smallest integer which is greater than i. DW3. N* = {i  I | 0  i}. The following corollary which is dual to Condition (1): A  I  i  A  i  Ij  A (j < i)  1k  Aj  A (j  k). (3) permits introducing the definition of maximum of a certain set of integers: DW4. A  I  i  A  i  Ij  A (j  i)  (k = max(A)  k  A j  A (j  k)). Let L(i) = {j I | j < i}. Hence L(i)   (see W'6) and 1k  L(i)j  L(i) (j  k) (see Condition (3)) then on the basis of DW4 the predecessor of integer i is defined as the greatest integer less than i, that is DW5. *i = max(L(i)), and set *N is defined as follows DW6. *N = {i  I | i  0}. 3.3. Equivalence of Systems P1IA and WIA Remark 3. With the definitions of the primitive notions of system P1IA, given in system WIA, all the axioms and definitions P1IA become theorems of definitions in system WIA. The definitions of addition and multiplication, which are accepted in P1IA are the same in system WIA, while definition DI4 of relation < accepted in system P1IA is a theorem in system WIA. Thus, it follows from Remarks 3 and 2 and MT8 that MT10. System WIA is equivalent to those of Iwanuś P1IA and P2IA. It follows from the above and MT9 that MT11. All the systems of integer arithmetic: WIA, P1IA, P2IA and SIA are mutually equivalent. In particular, MT12. System P1IA modelled on Peano's system PA and system WIA modelled on Wilkosz's system WA are equivalent. 3.4. Independence of the Axioms in P1IA and WIA The axioms of the integer arithmetic system P1IA can, as Iwanuś proved, be reduced by one axiom A*3 or *A3. If we found an axiom system of P1IA on those of A*1–A*4 and *A1, *A2 and *A4, then axiom *A3 can be proved. It follows from *1, A*2, and A5 as well as from theorems of this system: *  N* and i N*  (i  *N  i = 0). MT13. The set of axioms of system P1IA can be based on an independent set of axioms A*1– A*4 and *A1, *A2, *A4, and A5. The independence of these axioms was proved by interpretation in integer arithmetic IA. The primitive terms of the tuple <N*, *N, i*, *i, 0> correspond to the elements of a tuple in the form <A, B, f(i), G(i), a0>, respectively, which does not satisfy only one axiom of P1IA. When we apply the denotation: " + " denotes a set of non‐negative integers, * i ∈ *⇒ (i Axioms 2019, 8, x FOR PEER REVIEW 11 of 14 Condition (1) allows intro uci g correctly the def ition of minimum in set A: DW1. A  I  i  A  i  Ij  (i  j)  (k = min (A)  k  A  j  A (k  j)). It f llows from Condition (1) and DW1 that there is a u ique minimum, min (A), when A  I, A   and set A has a lower bound. Let G(i) = {j I | i < j}. Hence G(i)   (see W'5) and 1k G(i)j G(i) (k  j) (s e Condition (1)); then on the basis of DW1 1k  I (k = min (G(i)). (2) The successor f an integer i is introduced by means of the definition: DW2. i* = in(G(i)) – i* is the smallest integer which is greater than i. DW3. N* = {i  I | 0  i}. The following corollary wh ch is dual to Condition (1): A  I  i  A  i  Ij  A (j < i)  1k  Aj  A (j  k). (3) permits introducing the definition of maximum of a certain set of integers: DW4. A  I  i  A  i  Ij  (j  i)  (k = max(A)  k  A j  A (j  k)). Let L(i) = {j I | j < i}. Hence L(i)   (see W'6) and 1k  L(i)j  L( ) (j  k) (see Condition (3)) then on the basis of DW4 the predec ssor of integer i is defin d as the greatest integer less than i, that is DW5. *i = max(L(i)), and set *N is defined as follows DW6. *N = {i  I | i  0}. 3.3. Equivale ce of Systems P1IA and WIA Remark 3. With the definitions of the pri itive notions of system P1IA, given in system WIA, all the axioms and definitions P1IA become theorems of definitions in system WIA. The definitions of additio and multiplication, which are accepted in P1IA are the same in system WIA, while definition DI4 of relation < accepted in system P1IA is a theorem in system WIA. Thus, it follows from Remarks 3 and 2 and MT8 that MT10. System WIA is equivalent to those of Iwanuś P1IA and P2IA. It follows from t e above and MT9 that MT11. All the systems of integer arithmetic: W , P1IA, P2IA and SIA are mutually equivalent. In particular, MT12. System P1IA modelled on Peano's system PA and system WIA modelled on Wilkosz's system WA are equivalent. 3.4. Indepe dence of the Axioms in P1IA and WIA The axioms of the int ger arithmetic system P1IA can, as Iwanuś proved, be reduced by one axiom A*3 or *A3. If we found an axiom system of P1IA on thos of A*1–A*4 and *A1, *A2 and *A4, th n axiom *A3 can be proved. It f llows from A*1, A*2, and A5 as well as from theorems of this system: *0 N* and i N*  (i  *N  i = 0). MT13. The set of axioms of system P1IA can be based on an independent set of axioms A*1– A*4 and *A1, *A2, *A4, and A5. The independence of these axioms was pr ved by interpretation in integ r arithmetic IA. The primitive terms of the tuple <N*, *N, i*, *i, 0> correspond to the elements of a tuple in the form <A, B, f(i), G(i), a0>, respectively, which does not satisfy only on axiom of P1IA.When we apply the denotation: " + " d no es a set of non‐negative integers, * ∨ i ). . s t f axioms of system P1IA can be based on an i dep ndent set f axioms A*1–A*4 and *A1, *A2, *A4, and A5. i c f t s i s s r i t r r t ti i i t r rit tic I . ri iti t r s f t t l *, * , i*, *i, correspond to t e l ts f t l i t f r , , f(i), G(i), a0>, respectively, which do s not satisfy only one axiom of P1IA. When we apply the denotation: "N + " denotes a set of non-negative integers, "N − " denotes a set of non-positive integers, "E+ " denotes a set of even non-negative integers; "E− " denotes a set of even non-positive integers, then the tuple: <N + \ {0}, N -, i + 1, i – 1, 0> does not satisfy A*1, <N +, N - \ {0}, i + 1, i – 1, 0> does not satisfy *A1, <N + \ {1}, N -, i + 1, i 1, 0> does not satisfy A*2, <N +, N - \ {-1}, i +1, i 1, 0> does not satisfy *A2, <{0, 1}, {0, 1}, f1(i), g1(i), 0>, where f1(i) = g1(i ) = { 1 f or i=0 0 f or i,0 does not satisfy A*3, <N +, E-, i + 2, i – 2, 0> does not satisfy A*4, <E+, N -, i + 2, i – 2, 0> does not satisfy *A4, <{0, 1}, N-, f2(i), i – 1, 0>, where f2(i) = { i+1 f or i≤0 1 f or i>0 does not satisfy A5a, <N +, {0, 1}, i + 1, g2( i), 0>, where g2( i) = { i−1 f or 0≤i −1 f or i<0 does not satisfy A5b, when A5 is substituted by two axioms: A5a. i ∈ N* ∪ *N⇒ *(i*) = I; A5b. i ∈ N* ∪ *N⇒ (*i)* = i. It is also possible to reduce the system of the primitive notions of P1IA system by one primitive notion-zero 0-since the following expression: i = 0⇔ i ∈ N* ∧ i ∈*N is a theorem of P1IA. On the other hand, MT14. The set of axioms I1-I4 of Iwanuś's P2IA system is an independent set. It is so, since applying the following interpretation: 1. I→ N +, i*→ |i | +1, 0→ 0, – I1 is not satisfied, 2. I→ E + ∪ {1}, i*→ i + 1, if i , 0, and i*→ 1, for i = 1, 0→ 0 – I2 is not satisfied, 3. I→ {0,1}, i*→ 1 - |i |, 0→ 0, – I3 is not satisfied, Axioms 2019, 8, 103 12 of 14 4. I→ set of integers Z, i*→ i + 2, 0→ 0, – I4 is not satisfied. It can also be justified that MT15. The set of axioms of WIA system is an independent set. 3.5. Categoricity of the Axiomatic Systems of Integers Arithmetic IA The classical model of P2IA system is the triple <Z, *, 0>, where Z is the set of all integers. The classical model of WIA system is the triple <Z, 0, < >. It can be proved (cf. [11]) that MT16. Every two models of WIA system are isomorphic, therefore WIA system is categorical in power א0. A model of WIA system is every triple <θ, 0, ≺ > corresponding to that of <I, 0, < > of the primitive terms of WIA, in which θ is an infinite set of cardinality א0, 0 ∈ θ, and ≺ is a binary relation satisfying axioms A'1–A'8 of WIA system. The following theorem is true: If an axiomatic system has the property that all its models are isomorphic, then each equivalent system has the same property. Thus, from meta-theorems MT11 and MT16 follows the conclusion: MT17. All the systems of integer arithmetic, which are presented in this work, are categorical in power of א0. Thus, it is not only system WIA modelled on Wilkosz's system WA which is categorical, but also Iwanuś's system P1IA (P2IA) modelled on Peano's system PA is categorical. A separate proof that system SIA is also categorical is given in the book by Sierpiński [6]. All deductive systems of integer arithmetic presented in this paper have a standard model and all their models are isomorphic (MT17), so all the theorems of these systems are true. Hence, it follows that MT18. The systems P1IA, P2IA, SLA and WIA of integer arithmetic are consistent. All these systems are mutually equivalent. 4. Final Comments â Theorems of categoricity of the systems of natural numbers and the integer systems answer-in a sense-the following question: To what extent do our axioms characterize natural numbers (respectively, integers)? It follows from them that each set which has properties expressed in our axioms is the same as the set of natural numbers (resp., integers), that is it is isomorphic. The axioms given for systems PA and WA as well as, respectively, P1IA (P2IA) and WIA, characterize very strongly natural numbers (respectively, integers). â It follows from the given considerations that from the point of view of set theory, the set of axioms of integer arithmetic systems P1IA and P2IA, modelled on Peano's axioms of system PA of natural numbers arithmetic, and the set of axioms of system WIA modeled on Wilkosz's axioms of system WA of natural numbers arithmetic, have equal rights, similarly as the set of axioms of systems PA and WA. The subject of the discussion can be-as it may seem-solely one problem: Which of the set of axioms is more intuitive or more useful in the didactic process? Wilkosz's system WA and system WIA of the similar axiomatic character seem to be of certain greater value. The former (WA) can be acknowledged to arise as a result of studies of the natural model-one that forms the primary study of teaching in the early years of elementary school. As it appears, though, the problem of which of the systems discussed here can play its role in a better way as the curriculum of early education may be settled exclusively through psycho-sociological research in schools. Axioms 2019, 8, 103 13 of 14 â Let us also note that the built systems of integer arithmetic, modelled on the systems of arithmetic of natural numbers of Peano and Wilkosz, were treated as respective extensions of the latter, since the set of natural numbers N in Peano's system PA, with the function of successor * and zero 0 ∈ N, is isomorphic with a proper subset of set I of integers in the system of integer arithmetic P1IA, that is to set N*⊂ I, function * in N* and zero 0 ∈ N*, whereas the set of natural numbers N in Wilkosz's system of natural numbers arithmetic WA, with zero 0 ∈ N and relation less-than < in N, is isomorphic with the proper subset N* ⊂ I in the system of integers arithmetic WIA, with zero 0 ∈ N* and relation < in N*. â It follows from the remark above that arithmetic of integers can be defined not only through giving a set of axioms, but as an extension of arithmetic of natural numbers by the well-known method of construction, as well. â So, it follows from MT6 that both integer systems P1IA (P2IA) and WIA can be treated as fragments of set theory. Funding: This research received no external funding. Acknowledgments: I would like to express my utmost gratitude to all referees of this article for all remarks, comments, suggestions and efforts to accomplish improvement of the content of the paper and also English-language verification of its text. Individual thanks are due to Luna Shen for help in all electronic correspondence. Conflicts of Interest: The author declares no conflict of interest. References 1. Peano, G. Arithmetices Principia Nova Methodo Exposito; Bocca: Turin, Italy, 1889. 2. Wilkosz, W. Arytmetyka Liczb Całkowitych/System aksjomatyczny/(Arithmetic of Natural Numbers/The Axiomatic System/); Biblioteczka Kółka Mat. i Fiz. UJ. Nr 1: Kraków, Poland, 1932. 3. Iwanuś, B. O Pewnych Aksjomatykach Arytmetyki Liczb Całkowitych (Some Axiomatic Systems of Integers Arithmetic); Teacher's Training College Press: Opole, Poland, 1965; Matematyka; Volume 4, pp. 23–61. 4. Wybraniec, U. 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