To expand search, see Logic. Laterally related topics: Proof and Computation.
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Kilmister, C. W. Zeno, Aristotle, Weyl and Shuard: two-and-a-half millenia of worries over number. Math. Gaz. 64 (1980), no. 429, 149--158. (Reviewer: K. E. Hirst.) SC: 01A99 (00A05 03A05), MR: 82i:01075.
Ever since Zeno's paradoxes, mathematicians, philosophers, and logicians have been discussing the nature of the infinite. The author starts by discussing one of Zeno's four paradoxes, the Dichotomy. This leads to a discussion of Aristotle's views of the infinite. Needless to say, philosophical problems remained, and Hermann Weyl made one attempt to rectify them. Weyl advised caution in dealing with impredicative definitions, which he believed could lead to a vicious circle. Unfortunately, as Weyl notes "This vicious circle, which has crept into analysis through the foggy nature of the usual set and function concepts, is not a minor, easily avoided form of error in analysis." And in fact, if impredicative definitions are abandoned entirely, we must also abandon the notion that a bounded infinite set has a least upper bound and of course the related theorem (Bolzano-Wierestrass) that a bounded infinite set has a limit point. As the author notes, "On 9 February 1918, Polya and Weyl made a bet in Zürich, with twelve witnesses (all mathematicians). About [the least upper bound property], Weyl prophesied 'A. Within twenty years, Polya, or a majority of leading mathematicians, will admit that the concepts of number, set and countability involved are completely vague; and that there is no more point in asking about the truth of [the least upper bound property] than of the main assertions of Hegel's physics. B. It will be recognized by Polya, or a majority of leading mathematicians, that in any wording [the least upper bound property] is false...'" When the bet was called, everyone agreed that Polya had won with the single exception of Kurt Gödel. The author notes "if the construction of the real numbers contains subtleties that troubled such an acute intellect as Weyl's as recently as 1917, and still worried Gödel in 1940, it is not to be wondered at that some of our first-year undergraduates find it hard to stomach. Perhaps they are wiser than we are." Closely related topics: Zeno, Aristotle, Hermann Weyl, Paradox, and Philosophy.