History
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, Felix Hausdorff found a mistake in the proof. 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, in first order logic (The same applies to Zorn's Lemma.) . In second order logic, however, the well-ordering theorem is strictly stronger than the axiom of choice: from the well-ordering theorem one may deduce the axiom of choice, but from the axiom of choice one cannot deduce the well-ordering theorem.
Read more about this topic: Well-ordering Theorem
Famous quotes containing the word history:
“Every library should try to be complete on something, if it were only the history of pinheads.”
—Oliver Wendell Holmes, Sr. (1809–1894)
“I saw the Arab map.
It resembled a mare shuffling on,
dragging its history like saddlebags,
nearing its tomb and the pitch of hell.”
—Adonis [Ali Ahmed Said] (b. 1930)
“What has history to do with me? Mine is the first and only world! I want to report how I find the world. What others have told me about the world is a very small and incidental part of my experience. I have to judge the world, to measure things.”
—Ludwig Wittgenstein (1889–1951)