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Title: The Incomplete Universe: Totality, Knowledge, and Truth by Patrick Grim ISBN: 0-262-07134-7 Publisher: MIT Press Pub. Date: 08 November, 1991 Format: Hardcover Volumes: 1 List Price(USD): $33.00

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Average Customer Rating: 4 (1 review)

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Summary: Omniscience Abandoned
Comment: Goedel's 1931 incompleteness theorem proved that there can beno proof of the consistency of Peano's axioms of arithmetic within thesystem, and consequently demonstrated that there are arithmetical truths which do not lend themselves to formal proofs. As extraordinary and shocking as Goedel's result was, anyone who reads a detailed account of his theorem ( see for instance "Goedel's Proof" by Nagel and Newman) realizes that certain ingredients of Goedel's elaborate proof were already part of the dominant discourse of mathematical logic of his time. One such ingredient was Russell's Liar Paradox and its variants. Since the appearance of Cantor's set theory and Russell-Whitehead Principia Mathematica, the paradoxes such as the set of all sets, or the set of all sets which do not contain themselves as members have resulted in numerous contradictions and have caused alternative logics and set theories to be created. Re-examining the Liar, Grim attacks the problem of the totality of truth and the notion of omniscience. He shows time and again that the very concept of a collection of all truths is inherently incoherent. He also shows that omniscience as knowing and believing in all truths result in much the same contradictions. As such he illustrates with remarkable lucidity that the notion of a complete universe as one in which one can speak of all true statements should be abandoned. In each chapter of his book Grim proposes possible ways out of the dilemma and masterfully demonstrates that there are no ways out. No matter how we alter and improve our understanding and formalization of set theory ( for instance in different appeals to Class theories), and no matter how hard we try to revisit our basic logical assumptions ( for instance admitting multi-valent logics) one form of the Liar or another ( e.g. strngthened Liar) reappear. On rare occasions that there seem to be certain ways out the price and cost seem to be unreasonably high. We may have to abandon, in these rare case, our most intutive undrstanding of truth or certain important mathematical tools. Although some rudimentary familiarity with formal logic is assumed in Grim's book, the reader need not have much techincal competence mathematically. Grim tends to repeat his basic idea, that the notion of the totality of truth is inherently flawed, over and over again. So although the reader may find himself or herself bored with the repetition, he or she should not fear that the basic idea can possibly escape his or her attention ultimately. For people with minimal mathematical background and interesetd only in the purely philosophical aspects of the issue Chapter 3 of the book is particulary recommendable. There is a brilliant proof of what Grim calls Expressive Incompleteness which only pre-supposes the intutive understanding that a set is always smaller than the set of all its subsets. IF the reader can convince him or herself of this elementary fact of set theory--and life! --he or she should have no problem whatsoever to follow the argument of this chapter. Grim's book must appear very amusing to those interested in incompleteness results. It is written with simplicity and yet the simplicity has not cost the book its precision and rigor. Read it and enjoy. One may be tempted to call THE INCOMPLETE UNIVERSE blasphemous, but there is no doubt that it is nonetheless a heavy load of fun.