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Yourgrau P. A world without time.. the forgotten legacy of Goedel and Einstein (Basic Books, 2005)(ISBN 0465092934)(176s)_PPop_

Yourgrau P. A world without time.. the forgotten legacy of Goedel and Einstein (Basic Books, 2005)(ISBN 0465092934)(176s)_PPop_

Yourgrau P. A world without time.. the forgotten legacy of Goedel and Einstein (Basic Books, 2005)(ISBN 0465092934)(176s)_PPop_

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logical system, such as <strong>the</strong> one developed by Frege in Begriffsschrift, given its axioms <strong>and</strong><br />

pro<strong>of</strong> procedures, was both internally consistent (<strong>the</strong> axioms <strong>and</strong> pro<strong>of</strong> procedures could<br />

not be used to prove two statements that contradicted each o<strong>the</strong>r) <strong>and</strong> complete (<strong>the</strong><br />

pro<strong>of</strong> procedures sufficed to prove every true statement in <strong>the</strong> system). It was in<br />

answering just such questions that Godel discovered his famous incompleteness <strong>the</strong>orem.<br />

The development <strong>of</strong> ma<strong>the</strong>matical logic also went beyond Frege ins<strong>of</strong>ar as it replaced<br />

Frege's use <strong>of</strong> concepts (<strong>the</strong> Begriffe <strong>of</strong> his Begriffsschrift) with <strong>the</strong> extensions <strong>of</strong> <strong>the</strong>se<br />

conceptsói.e., <strong>the</strong> set <strong>of</strong> objects described by <strong>the</strong> conceptsówhich came to be known as<br />

sets or classes. But whereas for Frege <strong>the</strong> <strong>the</strong>ory <strong>of</strong> concepts <strong>and</strong> <strong>the</strong>ir extensions had<br />

been contained in logic itselfó<strong>the</strong> very part <strong>of</strong> Frege's <strong>the</strong>ory that Bertr<strong>and</strong> Russell would<br />

later show contained an inconsistencyóas <strong>the</strong> field developed in <strong>the</strong> early years <strong>of</strong> <strong>the</strong><br />

twentieth century, set <strong>the</strong>ory came into being as a new field unto itself, with its own<br />

axioms. This new axiomatic set <strong>the</strong>ory, developed by such thinkers as Ernst Zermelo<br />

<strong>and</strong> Abraham Fraenkel, replaced both <strong>the</strong> axiomatized <strong>the</strong>ory <strong>of</strong> concepts <strong>and</strong> <strong>the</strong>ir<br />

extensions <strong>of</strong> Frege, as well as <strong>the</strong> earlier, unaxioma-tized, "naive" set <strong>the</strong>ory <strong>of</strong> Cantor.<br />

Frege's work gave birth, <strong>the</strong>n, to two new subfields <strong>of</strong> ma<strong>the</strong>matics: ma<strong>the</strong>matical logic<br />

<strong>and</strong> axiomatic set <strong>the</strong>ory.<br />

Godel's ma<strong>the</strong>matical advisor, Hans Hahn, kept abreast <strong>of</strong> all <strong>the</strong>se new developments in<br />

ma<strong>the</strong>matical logic <strong>and</strong> set <strong>the</strong>ory. Indeed, he directed a seminar devoted to <strong>the</strong> classic <strong>of</strong><br />

modern ma<strong>the</strong>matical logic, Principia Matbematica, by Bertr<strong>and</strong> Russell <strong>and</strong> Albert North<br />

Whitehead. Godel did not participate in that seminar, but he did attend one given by <strong>the</strong><br />

philosopher Moritz Schlick on Bertr<strong>and</strong> Russell's later work, Introduction to Ma<strong>the</strong>matical<br />

Philosophy (written while Russell was in jail in Engl<strong>and</strong> for his protests against British<br />

participation in World War I). Godel also attended a seminar on <strong>the</strong> foundations <strong>of</strong><br />

ma<strong>the</strong>matics <strong>of</strong>fered by Rudolph Carnap, who had been a student <strong>of</strong> Frege's at <strong>the</strong><br />

University <strong>of</strong> Jena. Carnap would shortly become one <strong>of</strong> <strong>the</strong> most influential members <strong>of</strong><br />

Schlick's "Vienna Circle." In a city full <strong>of</strong> cliques, salons <strong>and</strong> discussion groups on every<br />

conceivable topic, <strong>the</strong> Vienna Circle was <strong>the</strong> most exclusive.<br />

Outside <strong>the</strong> circle, Godel's life was not unlike those <strong>of</strong> o<strong>the</strong>r well-<strong>of</strong>f Viennese<br />

intellectuals. With his bro<strong>the</strong>r, Rudolf, his senior by four years, Kurt lived in a comfortable<br />

apartment in which <strong>the</strong>y were joined regularly by <strong>the</strong>ir mo<strong>the</strong>r, Marianne. Toge<strong>the</strong>r with<br />

her, <strong>the</strong> two bro<strong>the</strong>rs enjoyed automotive vacations in Rudolf's new Chrysler, one <strong>of</strong> <strong>the</strong><br />

first in <strong>the</strong> region, to spots as far away as Marienbad. Though <strong>the</strong> family employed a<br />

chauffeur, on vacations <strong>the</strong> bro<strong>the</strong>rs preferred doing <strong>the</strong> driving. Kurt liked to drive fast.<br />

This, combined with his penchant for indulging in abstract reverie while behind <strong>the</strong> wheel,<br />

led his future wife, Adele, to put an end to his driving career. In town, Marianne made<br />

certain that her two academic sons did not neglect <strong>the</strong> full cultural <strong>of</strong>ferings laid before<br />

<strong>the</strong>m by beautiful Vienna. These included plays, with box seats at Max Reinhardt's famous<br />

Josefstadt Theater, <strong>and</strong> concerts, especially light opera, <strong>of</strong> which Godel was especially<br />

fond.<br />

Godel's exceptionally clear mind made him a much sought after intellectual companion<br />

among his fellow students. He was generous with his <strong>time</strong> <strong>and</strong> patient with his<br />

interlocutors. His friend Karl Menger writes that Godel "always grasped problematic points

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