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:
“The history of the world is the record of the weakness, frailty and death of public opinion.”
—Samuel Butler (1835–1902)
“I believe that history might be, and ought to be, taught in a new fashion so as to make the meaning of it as a process of evolution intelligible to the young.”
—Thomas Henry Huxley (1825–95)
“To a surprising extent the war-lords in shining armour, the apostles of the martial virtues, tend not to die fighting when the time comes. History is full of ignominious getaways by the great and famous.”
—George Orwell (1903–1950)