Peano Axioms
Updated: October 21, 2020
A natural number is any element of the set 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++ ≠ 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≠m, then n++ ≠ 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.