# On Certain Axiomatizations of Arithmetic of Natural and Integer Numbers

## Abstract

**:**

## 1. Introduction

**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.

**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**.

**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].

## 2. Two Simple Axiomatizations of NA

#### 2.1. Peano’s Axioms for **PA**

**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).

**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.

**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

**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).

- T(m) for m ∈N
- T(k) ⇒T(k*) for k ≥ m
- --------------------------------
- T(n) for all n ≥ m, n ∈ N

- for each m ∈N
- T(k) ⇒ T(m) for any k < m
- -----------------------------------
- T(n) for all n ∈ N

**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, ≤.

**PA**, since the following relations of implication hold:

**Pmax.**In any non-empty set of natural numbers for which there is an upper bound element, there is the greatest number. Symbolically:

**Pmin.**In any non-empty set of natural numbers, there is the least number. Symbolically:

**MT1.**The principles P5, Pmax and Pmin are mutually equivalent on the basis of the elementary theorems of system

**PA**.

**Remark.**

**PA**by means of relations of less than, <, or non-greater than, ≤, but the former is defined by means of the operation of addition +.

**PA**are the following:

- D3. m < n ⇔ ∃k ∈ N \ {0} (m + k = n),
- D4. m ≤ n ⇔ m = n ∨ m < n.

**PA**:

- D1a. m + 0 = 0,
- b. m + n* = (m + n)*.
- D2a. m·0 = 0,
- b. m·n* = m·n + m.

**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).

**, 0, 1, ≤ > is an ordered commutative semi-ring.**

^{.}#### 2.2. Wilkosz’s Axioms for System **WA**

**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.

**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**

- (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**.

- (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**.

**MT2.**The systems of

**PA**and

**WA**are equivalent.

#### 2.4. Independence of Axioms in Systems **PA** and **WA**

**PA**and

**WA**without depleting the set of theorems which can be proved about natural numbers.

**b.**The set of axioms of

**WA**system is dependent and can be reduced to the set:

- 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.

#### 2.5. Categoricity of Arithmetic Systems **PA** and **WA**

**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).

**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.

**model**of Wilkosz’s arithmetic system

**WA**is each tuple <N, 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**: P_{1}**=**<N_{1}, 0_{1}, S_{1}> and P_{2}= <N_{2}, 0_{2}, S_{2}> are isomorphic if and only if there is bijection f: N_{1}_{→}N_{2}such that f is homomorphism from P_{1}to P_{2}, that isf(0_{1}) = 0_{2}and f(S_{1}(m)) = S_{2}(f(m)) for any m ∈ N_{1}. - (m2)
- Two models of
**WA**: W_{1}**=**<N_{1}, Հ_{1}> and W_{2}= <N_{2}, Հ_{2}> are isomorphic if and only if there is bijection f: N_{1}_{→}N_{2}being homomorphism from W_{1}to W_{2}, that ism Հ_{1}n ⇒ f (m) Հ_{2}f(n) for any m, n ∈ N_{1}.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 - (m4)
- Each two models of arithmetic system
**WA**are isomorphic.

**MT4.**The deductive systems

**PA**and

**WA**of natural numbers arithmetic

**NA**are categorical; they are in power ℵ

_{0}, so they are aleph-null categorical systems.

#### 2.6. Set-Theoretical Models for **PA** and **WA**

**PA**is the triple < ℕ, card (∅), S* >,

**WA**is the triple < ℕ, ≺ >,

**MT5. PA**and

**WA**systems are consistent.

**MT6.**Systems

**PA**and

**WA**are (treated as) fragments of set theory.

## 3. Simple Axiomatizations of Arithmetic of Integers, Based on Systems PA and WA

**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

**IA**system. They are interesting due to their intuitive character. The first system based on them will be denoted as

**P**, and the other one—

^{1}IA**P**. The primitive notions of

^{2}IA**P**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

^{1}IA**=**i – 1.

#### 3.1.1. Axioms of System **P**^{1}IA Are the Symmetric Axioms for Numbers of the Sets N* and *N:

^{1}IA

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)*.

**IA**, the symmetrical axioms of

**P**can be replaced by weaker ones, deriving from Słupecki in [11]:

^{1}IA- 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,

- (i ∈ I ∧ i ≠ *0) ⇒ i* ≠ 0,
- (i ∈ I ∧ i ≠ 0*) ⇒ *i ≠ 0.

**P**, there are the following definitions of the operations of: addition +, subtraction − and multiplication:

^{1}IAD^{I}1 a. i + 0 = i, | D^{I}2 a. i – 0 = i, | D^{I}3a. 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. |

I1. *N = (I – N*) ∪ {0}, | I2. k = j – i ⇔ i + k = j, |

I3. i* = i + 1, | I4. *i = i –1. |

**P**the following meta-theorem is made use of:

^{1}IA**MT7.**If α is an expression of system

**P**, in which—beside primitive notions—there are exclusively the defined terms + and

^{1}IA**, then α is a theorem of this system if expression α**

^{.}^{d}, dual with respect to α, is a thesis of this system; expression α

^{d}is dual to α, when the terms:

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.**

**P**one can prove all the axioms of the commutative ring.

^{1}IA**P**:

^{1}IA- D
^{I}4. i < j ⇔ ∃k ∈ N*\{0} (i + k = j).

**Remark**

**2.**

**P**, one can prove all the theorems of arithmetic of integers

^{1}IA**IA**relating to relation <.

#### 3.1.2. The Other System of Arithmetic of Integers Built by Iwanuś [3] and Modelled on System **PA**

**P**and based only on the following three primitive notions:

^{2}IA**P**:

^{2}IA- (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.

**P**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:

^{2}IAI3a. N ⊆ I, |

b. 0 ∈ N, |

c. i ∈ N ⇒ i* ∈ N, |

d. i ∈ N ⇒ i* ≠ 0. |

**P**system the primitive notions of

^{2}IA**P**system are defined in the following way:

^{1}IA- D
^{I}1′. i, j ∈ I ⇒ (*i = j ⇔ i = j*), - D
^{I}2′. i ∈ N* ⇔ ∀A ⊆ I (0 ∈ A ∧∀j ∈A (j* ∈ A)) ⇒ i ∈ A, - D
^{I}3′. i ∈ *N ⇔ ∀A ⊆ I (0 ∈ A ∧∀j ∈A (*j ∈ A)) ⇒ i ∈ A.

**P**are the same in system

^{1}IA**P**.

^{2}IA**MT8**. Systems

**P**and

^{1}IA**P**are equivalent.

^{2}IA- 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).

**P**are introduced into system

^{1}IA**SIA**as follows:

- D
^{S}1. *N = (I\N*) ∪ {0}, - D
^{S}2. k = j – i ⇔ i + k =j, - D
^{S}3. i* = i + 1, - D
^{S}4. *i = i – 1.

^{I}4 of relation < of system

**P**is the same as in system

^{1}IA**SIA**.

**MT9.**System

**P**(

^{1}IA**P**system), modelled on Peano’s system of natural numbers arithmetic

^{2}IA**PA**, and system

**SIA**are equivalent.

#### 3.2. Axioms of the System of Integer Arithmetic **WIA** Modelled on Wilkosz’s System **WA**

**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):

**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),
- 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).

**WA**. The axioms of system

**WIA**state that relation < orders set I, yet do not state that it well-orders the set.

**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

**P**. Let us note first that in system

^{1}IA**WIA**it is possible to prove the theorem:

_{1}k ∈ A∀j ∈ A (k ≤ j).

^{W}1. A ⊆ I ∧ ∃i ∈ A ∧ ∃i ∈ I∀j ∈ A (i ≤ j) ⇒ (k = min (A) ⇔ k ∈ A ∧ ∀j ∈ A (k ≤ j)).

^{W}1 that there is a unique minimum, min (A), when A ⊆ I, A ≠ ∅ and set A has a lower bound.

_{1}k ∈G(i)∀j ∈G(i) (k ≤ j) (see Condition (1)); then on the basis of D

^{W}1

_{1}k ∈ I (k = min (G(i)).

^{W}2. i* = min(G(i)) – i* is the smallest integer which is greater than i.

^{W}3. N* = {i ∈ I | 0 ≤ i}.

_{1}k ∈ A∀j ∈ A (j ≤ k).

^{W}4. A ⊆ I ∧ ∃i ∈ A ∧ ∃i ∈ I∀j ∈ A (j ≤ i) ⇒ (k = max(A) ⇔ k ∈ A ∧∀j ∈ A (j ≤ k)).

_{1}k ∈ L(i)∀j ∈ L(i) (j ≤ k) (see Condition (3)) then on the basis of D

^{W}4 the predecessor of integer i is defined as the greatest integer less than i, that is

^{W}5. *i = max(L(i)),

^{W}6. *N = {i ∈ I | i ≤ 0}.

#### 3.3. Equivalence of Systems **P**^{1}IA and **WIA**

^{1}IA

**Remark**

**3.**

**P**, given in system

^{1}IA**WIA**, all the axioms and definitions

**P**become theorems of definitions in system

^{1}IA**WIA**.

**P**are the same in system

^{1}IA**WIA**, while definition D

^{I}4 of relation < accepted in system

**P**is a theorem in system

^{1}IA**WIA**.

**MT8**that

**MT10**. System

**WIA**is equivalent to those of Iwanuś

**P**and

^{1}IA**P**.

^{2}IA**MT9**that

**MT11**. All the systems of integer arithmetic:

**WIA**,

**P**and

^{1}IA, P^{2}IA**SIA**are mutually equivalent.

**MT12**. System

**P**modelled on Peano’s system

^{1}IA**PA**and system

**WIA**modelled on Wilkosz’s system

**WA**are equivalent.

#### 3.4. Independence of the Axioms in **P**^{1}IA and **WIA**

^{1}IA

**P**can, as Iwanuś proved, be reduced by one axiom A*3 or *A3. If we found an axiom system of

^{1}IA**P**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:

^{1}IA**MT13.**The set of axioms of system

**P**can be based on an independent set of axioms A*1–A*4 and *A1, *A2, *A4, and A5.

^{1}IA**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), a

^{0}>, respectively, which does not satisfy only one axiom of

**P**. When we apply the denotation:

^{1}IA- “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}, f
_{1}(i), g_{1}(i), 0>, where f_{1}(i) = g_{1}(i ) = ${\{}_{0fori\ne 0}^{1fori=0}$ 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
^{-}, f_{2}(i), i – 1, 0>, where f_{2}(i) = ${\{}_{1fori0}^{i+1fori\le 0}$ does not satisfy A5a, - <ℕ
^{+}, {0, 1}, i + 1, g_{2}( i), 0>, where g_{2}( i) = ${\{}_{-1fori0}^{i-1for0\le i}$ does not satisfy A5b,

A5a. i ∈ N* ∪ *N ⇒ *(i*) = I; | A5b. i ∈ N* ∪ *N ⇒ (*i)* = i. |

**P**system by one primitive notion—zero 0—since the following expression:

^{1}IA**P**.

^{1}IA**MT14.**The set of axioms I1—I4 of Iwanuś’s

**P**system is an independent set.

^{2}IA- I → ℕ
^{+}, i* → |i | +1, 0 → 0, – I1 is not satisfied, - I → E
^{+}∪ {1}, i*→ i + 1, if i ≠ 0, and i* → 1, for i = 1, 0 → 0 – I2 is not satisfied, - I → {0,1}, i* → 1 - |i |, 0→ 0, – I3 is not satisfied,
- I → set of integers ℤ, i* → i + 2, 0 → 0, – I4 is not satisfied.

**MT15.**The set of axioms of

**WIA**system is an independent set.

#### 3.5. Categoricity of the Axiomatic Systems of Integers Arithmetic IA

**P**system is the triple <ℤ, *, 0>, where ℤ is the set of all integers. The classical model of

^{2}IA**WIA**system is the triple <ℤ, 0, < >.

**MT16.**Every two models of

**WIA**system are isomorphic, therefore

**WIA**system is categorical in power ℵ

_{0}.

**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 ℵ

_{0}, 0 ∈ ϑ, and ≺ is a binary relation satisfying axioms A’1–A’8 of

**WIA**system.

**MT17.**All the systems of integer arithmetic, which are presented in this work, are categorical in power of ℵ

_{0}.

**WIA**modelled on Wilkosz’s system

**WA**which is categorical, but also Iwanuś’s system

**P**(

^{1}IA**P**) modelled on Peano’s system

^{2}IA**PA**is categorical.

**SIA**is also categorical is given in the book by Sierpiński [6].

**MT18.**The systems

**P**,

^{1}IA**P**,

^{2}IA**SLA**and

**WIA**of integer arithmetic are consistent.

## 4. Final Comments

- ➢
- 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.

**PA**and

**WA**as well as, respectively,

**P**(

^{1}IA**P**) and

^{2}IA**WIA**, characterize very strongly natural numbers (respectively, integers).

- ➢
- It follows from the given considerations that from the point of view of set theory, the set of axioms of integer arithmetic systems
**P**and^{1}IA**P**, modelled on Peano’s axioms of system^{2}IA**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?

**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.

- ➢
- 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**P**, 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^{1}IA**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*. - ➢
- 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.
- ➢
- So, it follows from MT6 that both integer systems
**P**(^{1}IA**P**and^{2}IA)**WIA**can be treated as fragments of set theory.

## Funding

## Acknowledgments

## Conflicts of Interest

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Wybraniec-Skardowska, U.
On Certain Axiomatizations of Arithmetic of Natural and Integer Numbers. *Axioms* **2019**, *8*, 103.
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Wybraniec-Skardowska U.
On Certain Axiomatizations of Arithmetic of Natural and Integer Numbers. *Axioms*. 2019; 8(3):103.
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