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

The same lesson was learned, the hard way, by David Hilbert, a lowering figure in
mathematics, who, inspired by Einstein, had formulated the equarioiis of general relativity
five days before Einstein himself
succeeded, a situation which led, unsurprisingly, to some uncomfortable moments in their
relationship. The positivistic creedóby its own nature as opposed to the spirit of
mathematics as to philosophyóhad in the course of time found a home in mathematics as
well. As the pos-itivists would have it, the hierarchy of transfinite numbers discovered by
Georg Cantor, a surprising consequence of his theory of sets, was cast into disrepute for
bearing the stain of Platonism, for pointing to infinite horizons beyond the frame of the
natural realm. The great Hilbert, however, defended Cantor's set theory, proclaiming, "No
one shall expel us from the paradise that Cantor has created," and calling it "one of the
supreme achievements of purely intellectual human activity."
Cantor's paradise was a lush tropical domain of infinities that he claimed to have
encountered at the very heart of mathematics. The importance of a sound theory of
infinity was lost on neither mathematicians nor physicists. Mathematics, a tool
indispensable to physicists, had been undergoing a gradual development of increased rigor
and clarification of foundations, a process that came to fruition in the second half of the
nineteenth century and the first years of the twentieth. Infinity played an essential role.
Once and for all, it seemed, a firm foundation had been laid for the calculus invented by
Newton and Leibniz, in which so-called infinitesimals (infinitely small quantities) enjoyed
an ambiguous twilight existence between finitude and infinity. Weierstrass, Cauchy, Cantor
and others developed the modern theory of limits of infinite sequences, which for the first
time made rigorous sense of Newtonian concepts like "point" and "instantaneous velocity."
Further, Cantor, Frege, Dedekind and others put forward a convincing theory of real
numbersórational numbers as infinite sequences of natural numbers, and irrational
numbers as infinite sequences of rational numbersówhich was crucial, since the physical
continuum of space and time could be fully described only by the real numbers. (Frege also
advanced an account of the natural numbers in terms of infinite aggregates of concepts,
but this fell on deaf ears.) All of this required, however, a comprehensive mathematical
theory of sequences, or more generally groupings, sets or classes of numbers, as well as a
mathematical account of infinity. Cantor, in a single bold move, developed precisely what
was needed, a set theory that provided a rigorous account of infinite sets.
His first discovery was that the requisite infinity had to be "actual," which went against a
two-thousand-year tradition in mathematics, from Aristotle to Gauss, which held that
infinity is merely "potential." Before Cantor, it was axiomatic that infinity was not to be
considered a definite number. To say, for example, that the natural numbers are infinite in
number was taken to mean not that there is an actual number, infinity, that numbers the
set of natural numbers, but rather that the set of natural numbers goes on forever, and
that the most that one can say is that no natural number is big enough to number the
entire set. Cantor, in contrast, produced a powerful argument for the thesis that there is
an actual number, which he called Xo (aleph null), that numbers the set of natural
numbers. Naturally, he emphasized a fact that we can put as follows: The number that
numbers the natural numbers cannot itself be a natural number. It must be an unnatural,
or supranatural, or (as Cantor characterized it) transfinite number. The king cannot arise
from the class of peasants. What established the significance of such a transfinite number