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

What Godel discovered, however, was that not only are the Peano postulates in fact
incomplete, any system of axioms or postulates (even if infinitely large) from which
arithmetic can be derived that satisfies any reasonable mathematical criteria of
surveyability by a finite mind is of necessity incomplete. (An infinite mind, like God's,
which can grasp all the numbers at once, presumably has no need of axioms.) So the
simplest and most basic domain of mathematics, the arithmetic of the natural numbers,
the rock on which the grand edifice of mathematics stands, turns out to be, from a formal
axiomatic point of view, incomplete, and even worse, incompletable. Indeed, since a
computer can prove only theorems based on the axioms its programmer has fed into itóit
cannot, as Godel emphasized, create new axioms on its ownóit follows that in principle no
computer or fully specified system of computers, even if infinite, will ever capture all the
truths of arithmetic (never mind the rest of mathematics). As Godel put it, "Continued
appeals to mathematical intuition are necessary . . . for the solution of the problems of
finitary number theory."
The mathematical fact of the incompleteness of formal arithmetic, moreover, is accessible
not only to us, thinkers with minds and mathematical intuitions; ironically, a computer can
be programmed to prove Godel's theorems, the very theorems that establish the intrinsic
limitations of computers. The truths of arithmetic, then, cannot in principle be confined to
a formal system. Here is a crucial difference between truth and proof: a mathematical
proof, in the sense in which we are discussing it here, is always a proof in, and relative to,
a given formal system, whereas truth, as such, is absolute. What Godel proved is that
mathematical truth is not reducible to (formal or mechanical) proof. Syntax cannot
supplant semantics. The leitmotif of the twentieth century, it turns out, stands in need of
revision. Mechanical rules cannot obviate the need for meaning, and what gives us access
to meaning,
namely, intuition, cannot be dispensed with even in mathematics, indeed, even in
arithmetic. This was the first nail in Hilbert's coffin.
The second nail was not long in coming. Godel soon proved his second incompleteness
theorem, which demonstrated, with yet further irony, that if a given system of axioms for
arithmetic were in fact consistent, then it could not be proved consistent by the system
itself. Put otherwise, only an inconsistent formal system can prove its own consistency!
Von Neumann, the quickest of the quick, having heard Godel announce his incompleteness
results, derived, shortly thereafter, the unprovability of consistency. "I would be very
much interested," he wrote Godel, "to hear your views on this. ... If you are interested, I
will send you the proof details." One can imagine his disappointment when Godel informed
him that the manuscript for the second theorem was already on its way to the editors. It
was Von Neumann, however, who argued, against Godel himself, that the unprovability of
consistency, as Godel had demonstrated it, left no wiggle room for the Hilbert program.
Whereas for several years, Godel was cautious not to prejudge the question of whether
Hilbert might discover a finitary proof of consistency to which Godel's second theorem did
not apply, Von Neumann, from the beginning, was confident that this could never happen.
Assuming that one rejects Russell's controversial "axiom of reducibility," he said, "one
cannot obtain a foundation for classical mathematics via logical means." Von Neumann's
striking prescience, however, concerning the full significance of what Godel had
discovered may well have served only to deepen his regret that he had not been the first
to make these discoveries. Even the fact that he was one of the fathers of the modern