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 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.
The standard ordering ≤ of the integers is not a well-ordering, since, for example, the set of negative integers does not contain a least element.
The standard ordering ≤ of the positive real numbers is not a well-ordering, since, for example, the open interval (0, 1) does not contain a least element.
There exists proofs depending on the axiom of choice that it is possible to wellorder the real numbers, but these proofs are non-constructive and no one has yet shown a method to wellorder the real numbers.
Georg Cantor considered the well-orderingtheorem to be a "fundamental principle of thought." Most mathematicians however find it difficult to visualize a well-ordering of, for example, the set
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