|
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 therorem to be a "fundamental
principle of thought." Most mathematicians however find it difficult to imagine that the set 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 "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.
See also well-ordering principle.
|