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Well-ordering theorem
From Wikipedia, the free encyclopedia
The well-ordering theorem (not to be confused with the well-ordering axiom) states that every set can be well-ordered.
This is important because it makes every set susceptible to the powerful technique of transfinite induction.
Georg Cantor considered the well-ordering theorem to be a "fundamental principle of thought." Most mathematicians however find it difficult to visualize a well-ordering of, for example, the set R of real numbers; in 1904, Gyula Kőnig claimed to have proven that such a well-ordering cannot exist. A few weeks later, though, Felix Hausdorff found a mistake in the proof. Ernst Zermelo then introduced the axiom of choice as an "unobjectionable logical principle" to prove the well-ordering theorem. It turned out though, that the well-ordering theorem is equivalent to the axiom of choice, in the sense that either one together with the Zermelo-Fraenkel axioms is sufficient to prove the other. (Incidentally, the same applies to Zorn's Lemma.)
The well-ordering theorem has consequences that may seem paradoxical, such as the Banach–Tarski paradox.