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.