1Preface072710

1Preface072710 - 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 statements of an abstract set theoretic nature have been shown to be undecided in ZFC, starting with the pioneering work of Kurt Gödel and Paul J. Cohen. Some of these statements now known to be undecided in ZFC were not, at first, generally recognized to be of an abstract set theoretic nature - although they are today. 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 mathematical 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 that was highly non concrete, and out of touch with normal mathematical activity. According to conventional wisdom, reasonably well motivated
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1Preface072710 - 1 PREFACE The standard axiomatization of...

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