well-ordering principle implies axiom of choice


Let C be a collectionMathworldPlanetmath of nonempty sets. Then SCS is a set. By the well-ordering principle, SCS is well-ordered under some relationMathworldPlanetmath <. Since each S is a nonempty subset of SCS, each S has a least member mS with respect to the relation <.

Define f:CSCS by f(S)=mS. Then f is a choice function. Hence, the axiom of choice holds. ∎

Title well-ordering principle implies axiom of choice
Canonical name WellorderingPrincipleImpliesAxiomOfChoice
Date of creation 2013-03-22 16:07:46
Last modified on 2013-03-22 16:07:46
Owner Wkbj79 (1863)
Last modified by Wkbj79 (1863)
Numerical id 7
Author Wkbj79 (1863)
Entry type Theorem
Classification msc 03E25
Related topic AxiomOfChoice
Related topic ZermelosWellOrderingTheorem