# Well-ordering theorem

## Theoretic principle in mathematics stating every set can be well-ordered. / From Wikipedia, the free encyclopedia

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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.