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

Hilbert soon recovered from the shock of GodePs discovery and proceeded to incorporate
and develop it in his and Bernays's new textbook on mathematical logic. The same cannot
be said, however, for Ernst Zermelo, the mathematician who had inaugurated the
axiomatic development of set theory after Russell's paradox had demonstrated that the
naive set theory developed by Cantor lacked a coherent philosophical foundation. Even
today, the axioms of Zermelo-Fraenkel set theory are the most widely used and accepted
in the field. Yet Zermelo, from beginning to end, was unable to understand or accept
GodePs results. He became their principal mathematical opponent. (Wittgenstein bears the
honor of being their chief philosophical detractor.)
Zermelo's difficulties were understandable. GodePs theorems traded on crucial distinctions
such as truth versus proof, semantics versus syntax, and completeness versus formal
consistency, distinctions that, though in the air, became fully clarified for the first time
only after GodePs proofs had appeared. It was not that Hilbert, the founder of formalism,
distinguished carefully between truth and proof and simply opted for the latter. Rather, as
Godel himself put the matter years later, "formalists considered formal demonstrability to
be an analysis of the concept of mathematical truth and, therefore, were of course not in
a position to distinguish the two." In the realm of mathematics, proof, for the
formalist, was indistinguishable from truth, and so any attempt to draw distinctions
between them was simply incomprehensible. Zermelo's philosophical framework, in turn,
though different from Hilbert's, was so contrary to GodePs that reconciliation was
impossible.
Fate brought the two men together at another mathematical meeting, this time in Bad
Elster, a year after the conference at Konigsberg. When it was suggested to Zermelo after
the talks were over that he meet with Godel for lunch on a nearby hill, he demurred,
complaining first that he "did not like GodePs looks," then that the supply of food was
insufficient, and finally that the climb would defeat him. Zermelo should have trusted his
instincts. He was finally talked into meeting with Godel, but the encounter, though polite,
was fruitless. He would soon write to Godel that he had a discovered a "major gap" in his
argument, and a lengthy replyórunning to ten handwritten pagesóby Godel did little to
disabuse him of his doubts. Having once failed to enlighten Zermelo, Godel apparently
gave it up as a lost cause, declining to respond even when Zermelo published his criticisms.
Carnap, when shown Zermelo's letters, agreed that he had "completely misunderstood"
GodePs achievement.
If Zermelo's intransigence was to be expected, Bertrand Russell's ambivalence was not. The
coauthor of the monumental Principia Mathematical, which provided the actual formal
system for GodePs proof, continued, late in life, to refer to GodePs results only guardedly.
In a letter written in 1963, Russell, while acknowledging the greatness of Godel's
achievement, did not conceal that he remained puzzled by it, asking rhetorically "are we
to think that 2 + 2 is not 4, but 4.001 ?" This suggests that Godel had purported to have
demonstrated a flaw in classical mathematics, which precisely misses the point of Godel's
theorem. Russell knew, however, that he had not yet fully thought this through. He
commented, dryly, that he was "glad [I] was no longer working at mathematical logic."
(Apparently, Godel was too. In a letter to a colleague he wrote that "Russell evidently
misinterprets my result; however, he does so in a very interesting manner. ... In
contradistinction