Editing Well-ordering theorem (section)

{{short description|Theoretic principle in mathematics stating every set can be well-ordered.}}
{{redirect|Zermelo's theorem|Zermelo's theorem in game theory|Zermelo's theorem (game theory)}}
{{distinguish|Well-ordering principle}}

In [[mathematics]], the '''well-ordering theorem''', also known as '''Zermelo's theorem''', states that every [[Set (mathematics)|set]] can be [[well-order]]ed. A set ''X'' is ''well-ordered'' by a [[strict total order]] if every non-empty subset of ''X'' has a [[least element]] under the ordering. The well-ordering 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 {{section link|Axiom of choice|Equivalents}}).<ref>{{cite book |url=https://books.google.com/books?id=rqqvbKOC4c8C&pg=PA14 |title=An introduction to the theory of functional equations and inequalities |page=14 |location=Berlin |publisher=Springer |isbn=978-3-7643-8748-8 |first=Marek |last=Kuczma |year=2009 |authorlink=Marek Kuczma}}</ref><ref>{{cite book |url=https://books.google.com/books?id=ewIaZqqm46oC&pg=PA458 |title=Encyclopaedia of Mathematics: Supplement |first=Michiel |last=Hazewinkel |year=2001 |authorlink=Michiel Hazewinkel |page=458 |location=Berlin |publisher=Springer |isbn=1-4020-0198-3 }}</ref> [[Ernst Zermelo]] introduced the axiom of choice as an "unobjectionable logical principle" to prove the well-ordering theorem.<ref name = "zer">{{cite book |url=https://books.google.com/books?id=RkepDgAAQBAJ&pg=PA23 |title=Handbook of Mathematics |first=Vialar |last=Thierry |year=1945 |page=23 |location=Norderstedt |publisher=Springer |isbn=978-2-95-519901-5 }}</ref> One can conclude from the well-ordering theorem that every set is susceptible to [[transfinite induction]], which is considered by mathematicians to be a powerful technique.<ref name = "zer"/> One famous consequence of the theorem is the [[Banach–Tarski paradox]].