Well-ordering Theorem

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 Number s; in 1904 , Julius König 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.

The well-ordering theorem has consequences that may seem paradoxical, such as the Banach–Tarski Paradox .

SEE ALSO

Well-ordering Principle

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Well-ordering Theorem