If we want fully understand numbers and number sets, we must learn how to construct each sets of numbers.

## Construction of number systems: Natural numbers

## Peano Axioms

The set of natural numbers is axiomatic introduced by Italian mathematician Giuseppe Peano. These axioms are known as Peano’s axioms.

**Peano’s Axioms **for **natural numbers**:

is a set with the following properties:

(1) has a distinguished element which we call

(2) There exists

(3) is one-to-one function (injective function)

(4) There does not exists an element such that (not surjective function)

(5) If such that: (Principal of Induction)

(a),

(b) if ,

then set .

A set is the set of natural numbers and its element are natural numbers. A function is called a **successor function**.

**Lemma 1**. If and , then there exists such that .

One of the main features of a set is that each element in it, except number , has an immediate predecessor and each element has an immediate follower. There is the smallest natural number, number , however, there is no the largest natural number.

Now, we can define operation of addition and multiplaying by recursive rules.

**Addition of natural numbers **

There is a unique function , , with the following properties:

,

.

**Theorem 1**. (*Associativity*) For all three natural numbers and the following is valid:

*Proof*.

Let be a set of all natural numbers for which is valid , that is:

We must prove that . Proof we conduct by using the principle of induction by .

Firstly, we must show that , that is :

It follows that .

Now, suppose that , that is . We must show that is also in the set , that is

We have

Therefore, .

Since and from the assumption that follows , by the principle of induction we conclude . The statement is true for all natural numbers and .

**Theorem 2**. (*Commutativity*) For any two natural numbers and the following is valid:

For proof the commutativity of natural numbers we need to impose two minor lemma.

**Lemma 2**. For any two natural numbers and is valid:

**Lemma 3**. For all natural number is valid:

*Proof*. ( **Theorem 2.**)

Proof we conduct by using the principle of induction by . We define a set as:

By the 5. Peano’s axiom we need to prove:

(1) :

Therefore, .

(2) Suppose that . Then , that is :

We have proven that .

Since and from the assumption that follows , by the principle of induction we conclude . The statement is true for all natural numbers and .

**Multiplication of natural numbers**

There is a unique function , , with the following properties:

, ,

, .

**The properties of multiplication of natural numbers**.

**Associativity**. For all three natural numbers and is valid:

**Commutativity**. For all two natural numbers and is valid:

**Distributive law**. For all natural numbers and is valid:

**Ordering on**

**Definition**. Let . We say that if there exists a such that x + z = y.

## Construction of number systems: Integers

Now, we can construct integers. Of course, we will use natural number to construct integers.

Consider the set , and relations if is valid . Let the set be equivalence classes under this relation. Now we can define on set like:

If sets and , then sets and are non-empty subsets of set and thus we may pick elements form set , and elements (c,d) from set .

Now we can define an operation (*addition*) denoted by .

Now, we want to define *multiplication of integers* denoted by like:

.

For natural numbers and integers we can write down corresponding properties.

- Associativity of addition:
- Commutativity od addition:
- Additive inverse:
- Existing of neutral element for addition:
- Distributivity:
- Associative of multiplication:
- Commutativity of multiplication:
- Existing of neutral element for multiplication: