1Preface050511

1Preface050511 - 1 PREFACE The standard axiomatization of...

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1 PREFACE The standard axiomatization of mathematics is given by the formal system ZFC, which is read "Zermelo Frankel set theory with the axiom of choice". The vast majority of mathematical proofs fit easily into the ZFC formalism. ZFC has stood the test of time. However, a long list of mathematically natural statements of an abstract set theoretic nature have been shown to be undecided (neither provable nor refutable) in ZFC, starting with the pioneering work of Kurt Gödel and Paul J. Cohen concerning Cantor's continuum hypothesis. Yet these statements involve general notions that are uncharacteristic of normal mathematical statements. The unprovability and unrefutability from ZFC depends on this uncharacteristic generality. For example, if we remove this uncharacteristic generality from Cantor's continuum hypothesis, we obtain a well known theorem of Aleksandrov and Hausdorff (see [Al16] and [Hau16]). Already as a student at MIT in the mid 1960s, I recognized the critical issue of whether ZFC suffices to prove or refute all concrete mathematically natural statements. Here concreteness refers to the lack of involvement of objects of a distinctly pathological nature. In particular, the finite, the discrete, and the continuous (on nice spaces) are generally considered concrete - although, generally speaking, only the finite is beyond reproach. From my discussions then with faculty and fellow students, it became clear that according to conventional wisdom, the Incompleteness Phenomena was confined to questions of an inherently set theoretic nature. The incompleteness would not appear if this uncharacteristic generality is removed. According to conventional wisdom, reasonably well motivated
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This note was uploaded on 08/05/2011 for the course MATH 366 taught by Professor Joshua during the Fall '08 term at Ohio State.

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1Preface050511 - 1 PREFACE The standard axiomatization of...

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