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

logical system, such as the one developed by Frege in Begriffsschrift, given its axioms and
proof procedures, was both internally consistent (the axioms and proof procedures could
not be used to prove two statements that contradicted each other) and complete (the
proof procedures sufficed to prove every true statement in the system). It was in
answering just such questions that Godel discovered his famous incompleteness theorem.
The development of mathematical logic also went beyond Frege insofar as it replaced
Frege's use of concepts (the Begriffe of his Begriffsschrift) with the extensions of these
conceptsói.e., the set of objects described by the conceptsówhich came to be known as
sets or classes. But whereas for Frege the theory of concepts and their extensions had
been contained in logic itselfóthe very part of Frege's theory that Bertrand Russell would
later show contained an inconsistencyóas the field developed in the early years of the
twentieth century, set theory came into being as a new field unto itself, with its own
axioms. This new axiomatic set theory, developed by such thinkers as Ernst Zermelo
and Abraham Fraenkel, replaced both the axiomatized theory of concepts and their
extensions of Frege, as well as the earlier, unaxioma-tized, "naive" set theory of Cantor.
Frege's work gave birth, then, to two new subfields of mathematics: mathematical logic
and axiomatic set theory.
Godel's mathematical advisor, Hans Hahn, kept abreast of all these new developments in
mathematical logic and set theory. Indeed, he directed a seminar devoted to the classic of
modern mathematical logic, Principia Matbematica, by Bertrand Russell and Albert North
Whitehead. Godel did not participate in that seminar, but he did attend one given by the
philosopher Moritz Schlick on Bertrand Russell's later work, Introduction to Mathematical
Philosophy (written while Russell was in jail in England for his protests against British
participation in World War I). Godel also attended a seminar on the foundations of
mathematics offered by Rudolph Carnap, who had been a student of Frege's at the
University of Jena. Carnap would shortly become one of the most influential members of
Schlick's "Vienna Circle." In a city full of cliques, salons and discussion groups on every
conceivable topic, the Vienna Circle was the most exclusive.
Outside the circle, Godel's life was not unlike those of other well-off Viennese
intellectuals. With his brother, Rudolf, his senior by four years, Kurt lived in a comfortable
apartment in which they were joined regularly by their mother, Marianne. Together with
her, the two brothers enjoyed automotive vacations in Rudolf's new Chrysler, one of the
first in the region, to spots as far away as Marienbad. Though the family employed a
chauffeur, on vacations the brothers preferred doing the driving. Kurt liked to drive fast.
This, combined with his penchant for indulging in abstract reverie while behind the wheel,
led his future wife, Adele, to put an end to his driving career. In town, Marianne made
certain that her two academic sons did not neglect the full cultural offerings laid before
them by beautiful Vienna. These included plays, with box seats at Max Reinhardt's famous
Josefstadt Theater, and concerts, especially light opera, of which Godel was especially
fond.
Godel's exceptionally clear mind made him a much sought after intellectual companion
among his fellow students. He was generous with his time and patient with his
interlocutors. His friend Karl Menger writes that Godel "always grasped problematic points