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Peano Axioms

Peano Axioms

Updated: October 21, 2020
Math Real-Analysis

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 nm, 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.