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 $a\le b$ 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
Canonical name	WellorderingPrincipleForNaturalNumbers
Date of creation	2013-03-22 11:46:38
Last modified on	2013-03-22 11:46:38
Owner	CWoo (3771)
Last modified by	CWoo (3771)
Numerical id	18
Author	CWoo (3771)
Entry type	Axiom
Classification	msc 06F25
Classification	msc 65A05
Classification	msc 11Y70
Related topic	MaximalityPrinciple
Related topic	WellOrderedSet
Related topic	ExistenceAndUniquenessOfTheGcdOfTwoIntegers