The well ordering theorem states that every set can be well ordered. Thus the set can be given a total order in which every nonempty subset has a least element. Over Zermelo-Fraenkel set theory without choice, this theorem is equivalent to the axiom of choice (Mendelson 1997, p. 275). Zermelo (1904, p. 516) proved it using the axiom of choice.
Well Ordering Theorem
See also
Axiom of Choice, Well Ordered Set, Well Ordering PrincipleExplore with Wolfram|Alpha
References
Mendelson, E. Introduction to Mathematical Logic, 4th ed. London, England: Chapman & Hall, 1997.Zermelo, E. "Beweis, daß jede Menge wohlgeordnet werden kann." Math. Ann. 59, 514-516, 1904. https://doi.org/10.1007/BF01445300.Cite this as:
Weisstein, Eric W. "Well Ordering Theorem." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/WellOrderingTheorem.html