In mathematics, the wellordering theorem, also known as Zermelo's theorem, states that every set can be wellordered. A set X is wellordered by a strict total order if every nonempty subset of X has a least element under the ordering. The wellordering theorem together with Zorn's lemma are the most important mathematical statements that are equivalent to the axiom of choice (often called AC, see also Axiom of choice § Equivalents).^{[1]}^{[2]} Ernst Zermelo introduced the axiom of choice as an "unobjectionable logical principle" to prove the wellordering theorem.^{[3]} One can conclude from the wellordering theorem that every set is susceptible to transfinite induction, which is considered by mathematicians to be a powerful technique.^{[3]} One famous consequence of the theorem is the Banach–Tarski paradox.
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Elementary Number Theory: WellOrdering Principle

1.3.1 Well Ordering Principle 1: Video

The Wellordering Principle and Mathematical Induction
Transcription
History
Georg Cantor considered the wellordering theorem to be a "fundamental principle of thought".^{[4]} However, it is considered difficult or even impossible to visualize a wellordering of ; such a visualization would have to incorporate the axiom of choice.^{[5]} In 1904, Gyula Kőnig claimed to have proven that such a wellordering cannot exist. A few weeks later, Felix Hausdorff found a mistake in the proof.^{[6]} It turned out, though, that in firstorder logic the wellordering theorem is equivalent to the axiom of choice, in the sense that the Zermelo–Fraenkel axioms with the axiom of choice included are sufficient to prove the wellordering theorem, and conversely, the Zermelo–Fraenkel axioms without the axiom of choice but with the wellordering theorem included are sufficient to prove the axiom of choice. (The same applies to Zorn's lemma.) In secondorder logic, however, the wellordering theorem is strictly stronger than the axiom of choice: from the wellordering theorem one may deduce the axiom of choice, but from the axiom of choice one cannot deduce the wellordering theorem.^{[7]}
There is a wellknown joke about the three statements, and their relative amenability to intuition:
The axiom of choice is obviously true, the wellordering principle obviously false, and who can tell about Zorn's lemma?^{[8]}
Proof from axiom of choice
The wellordering theorem follows from the axiom of choice as follows.^{[9]}
Let the set we are trying to wellorder be , and let be a choice function for the family of nonempty subsets of . For every ordinal , define an element that is in by setting if this complement is nonempty, or leave undefined if it is. That is, is chosen from the set of elements of that have not yet been assigned a place in the ordering (or undefined if the entirety of has been successfully enumerated). Then the order on defined by if and only if (in the usual wellorder of the ordinals) is a wellorder of as desired, of order type .
Proof of axiom of choice
The axiom of choice can be proven from the wellordering theorem as follows.
 To make a choice function for a collection of nonempty sets, , take the union of the sets in and call it . There exists a wellordering of ; let be such an ordering. The function that to each set of associates the smallest element of , as ordered by (the restriction to of) , is a choice function for the collection .
An essential point of this proof is that it involves only a single arbitrary choice, that of ; applying the wellordering theorem to each member of separately would not work, since the theorem only asserts the existence of a wellordering, and choosing for each a wellordering would require just as many choices as simply choosing an element from each . Particularly, if contains uncountably many sets, making all uncountably many choices is not allowed under the axioms of ZermeloFraenkel set theory without the axiom of choice.
Notes
 ^ Kuczma, Marek (2009). An introduction to the theory of functional equations and inequalities. Berlin: Springer. p. 14. ISBN 9783764387488.
 ^ Hazewinkel, Michiel (2001). Encyclopaedia of Mathematics: Supplement. Berlin: Springer. p. 458. ISBN 1402001983.
 ^ ^{a} ^{b} Thierry, Vialar (1945). Handbook of Mathematics. Norderstedt: Springer. p. 23. ISBN 9782955199015.
 ^ Georg Cantor (1883), “Ueber unendliche, lineare Punktmannichfaltigkeiten”, Mathematische Annalen 21, pp. 545–591.
 ^ Sheppard, Barnaby (2014). The Logic of Infinity. Cambridge University Press. p. 174. ISBN 9781107058316.
 ^ Plotkin, J. M. (2005), "Introduction to "The Concept of Power in Set Theory"", Hausdorff on Ordered Sets, History of Mathematics, vol. 25, American Mathematical Society, pp. 23–30, ISBN 9780821890516
 ^ Shapiro, Stewart (1991). Foundations Without Foundationalism: A Case for SecondOrder Logic. New York: Oxford University Press. ISBN 0198533918.
 ^ Krantz, Steven G. (2002), "The Axiom of Choice", in Krantz, Steven G. (ed.), Handbook of Logic and Proof Techniques for Computer Science, Birkhäuser Boston, pp. 121–126, doi:10.1007/9781461201151_9, ISBN 9781461201151
 ^ Jech, Thomas (2002). Set Theory (Third Millennium Edition). Springer. p. 48. ISBN 9783540440857.