Gödel's theorem : an incomplete guide to its use and abuse /

Godel's theorem is a landmark in the development of mathematics. It addresses a question posed by David Hilbert in his famous Paris address: Can we know that a system of axioms that describes arithmetic is consistent, that is, can we know that we will never prove from these axioms both a statement and its negation? An answer to this question ought to be a mathematical proof of the system's consistency. Godel's theorem tells us that no such proof is possible. He accomplished this remarkable result by showing the existence of statements in arithmetic that assert their own lack of proof by a version of the liar's paradox, "this sentence is a lie." The theorem has attracted all sorts of attention from fields outside mathematics. For example, it is sometimes used to explore questions in the philosophy of mind and in the abstract theory of computability. Goldstein discusses Godel's work in its historical framework--the Vienna of the 1920s, where the philosophical school of logical positivism was born--and in its personal aspect, the mathematician's life. The first and last chapters give rich descriptions of the Institute for Advanced Study, where Godel and Einstein--friends in exile--spent most of their careers. Her account of Vienna is lively, though her view of its influence on Godel's early work is speculative. Her gift as a novelist makes this account a pleasure to read. An excellent addition to the Norton series "Great Discoveries." Franzen's goal is different and ambitious. The use of Godel's work outside of mathematics often involves assumptions about the compass of the results that are in error. He discusses nonmathematical ways in which the incompleteness theorem has been invoked, e.g., in informal systems like the Bible or the Constitution. He also discusses computability, a field of study that began with Turing's work to recast the incompleteness theorem. Though Franzen makes more demands of his readers than does Goldstein, he is thorough and lively, and patient reading is repaid with fascinating results. Both books offer readers an opportunity to engage in Godel's profound insight and taste some of the implications it has when appropriately applied. Both are welcome additions to most libraries. ^BSumming Up: Both books: Highly recommended. General readers; lower-division undergraduates through faculty. J. McCleary Vassar College