well-ordering principle for natural numbers

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

Beware that there is another statement (which is equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath to the axiom of choiceMathworldPlanetmath) 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