When talking about natural numbers, an axiomatic definition for natural numbers is necessary. To do this, we will show how to define the basic operations for natural numbers. Peano axioms Initially we will axiomatically define the set $\mathbb{N}$ of natural numbers. To describe such properties, we will use the symbols 0 (zero) and s (successor). (a) $0 \in \mathbb{N};$ Zero belongs to the set of natural numbers. (b) $\forall n \in \mathbb{N}\quad s(n) \in \mathbb{N};$ For every number belonging to the set of naturals, the successor belongs to the natural ones. (c) $\forall n \in \mathbb{N}\quad s(n) \neq 0;$ For every number belonging to the set of natural numbers, the successor is different from zero. (d) $\forall m,n \in \mathbb{N}\quad s(n)=s(m) \Rightarrow n=m;$ For every two pairs of numbers belonging to the natural set, such that the successors are equal, this implies that these numbers are equal. (e) Let $P(x)$ be a property on elements of $\mathbb{N}.$ Apply: (i) $P(0);$ (ii) $\forall n \in \mathbb{N}, \quad P(n)\Rightarrow P(s(n))$ Then it is valid $P(n)$ for all $n \in \mathbb{N}.$ The axiom $(e)$ is known as the principle of induction. The set $\\{0, 1, 2, \dots\\}$ satisfies the above properties if we interpret the symbol $s(n)$ as $n + 1.$ To note that the principle of induction holds in such a set, suppose that it does not hold. Then there is $n$ the smallest such that is not worth P(n).