Well-ordering theorem

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 imagine that the set $\mathbb {R}$ of real numbers, for example, can be well-ordered; in 1904, Julius König claimed to have proven that they cannot be. 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.