# well-ordering principle for natural numbers

Every nonempty set $S$ of natural numbers contains a least element; that is, there is some number $a$ in $S$ such that $a\leq b$ for all $b$ belonging to $S$.

Beware that there is another statement (which is equivalent to the axiom of choice) called the well-ordering principle. It asserts that every set can be well-ordered.

Note that the well-ordering principle for natural numbers is equivalent to the principle of mathematical induction (or, the principle of finite induction).

Title well-ordering principle for natural numbers WellorderingPrincipleForNaturalNumbers 2013-03-22 11:46:38 2013-03-22 11:46:38 CWoo (3771) CWoo (3771) 18 CWoo (3771) Axiom msc 06F25 msc 65A05 msc 11Y70 MaximalityPrinciple WellOrderedSet ExistenceAndUniquenessOfTheGcdOfTwoIntegers