Peano Axioms

A natural number is any element of the set $\mathbb{N} = \{0, 1, 2, 3…\}$

I

0 is a natural number.

II

If $n$ is a natural number, then n++ is also a natural number.

III

0 is not the successor of any natural number; i.e., we have n++ $\ne$ 0 for every natural number n.

IV

Different natural numbers must have different successors; i.e., if $n$ and $m$ are natural numbers and $n \ne m$, then n++ $\ne$ m++. Equivalently, if n++ = m++, then we must have n = m.

V

(Principle of mathematical induction). Let $P(n)$ be any property pertaining to a natural number n. Suppose that $P(0)$ is true, and suppose that whenever $P(n)$ is true, $P(n++)$ is also true. Then $P(n)$ is true for every natural number n.