Well-ordering theorem

"Zermelo's theorem" redirects here. For Zermelo's theorem in game theory, see Zermelo's theorem (game theory).
Not to be confused with Well-ordering principle.

In mathematics, the well-ordering theorem, also known as Zermelo's theorem, states that every set can be well-ordered. 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 Axiom of choice § Equivalents).[1][2] Ernst Zermelo introduced the axiom of choice as an "unobjectionable logical principle" to prove the well-ordering theorem.[3] 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.[3] One famous consequence of the theorem is the Banach–Tarski paradox.

  1. ^ Kuczma, Marek (2009). An introduction to the theory of functional equations and inequalities. Berlin: Springer. p. 14. ISBN 978-3-7643-8748-8.
  2. ^ Hazewinkel, Michiel (2001). Encyclopaedia of Mathematics: Supplement. Berlin: Springer. p. 458. ISBN 1-4020-0198-3.
  3. ^ a b Thierry, Vialar (1945). Handbook of Mathematics. Norderstedt: Springer. p. 23. ISBN 978-2-95-519901-5.