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

property is empty. As things turned out, however, this was by no means the worst that
could happen. Russell, annoyingly, asked us to consider the property of being a set that is
not a member of itself. The set of small sets, for example, is not a member of itself (since
it is clearly not a small set), whereas the set of big sets surely is. Russell was able to show,
however, that there could not be such a thing as the set of all sets that are not members
of themselves. If there were such a set, it would have to be and also not be a member of
itself. It follows that it is not true that every property determines the set of things that
have that property. But then, which properties do determine sets, and more generally,
exactly which sets actually exist?
Russell's paradox was disarmingly simple. It left mathematicians breathless. How, one
wonders, did Russell ever come up with his dangerous idea? Historical research has
revealed that he invented his paradox in the course of trying to refute Cantor's proof,
rehearsed above, that there are more real numbers than natural numbers. His arrow
missed Cantor but struck Frege squarely in the chest, toppling his formal development of
set theory and shattering his life's work. We still have the polite and lethal letter Russell
sent to Frege in 1902: "Dear Colleague, ... I find myself in agreement with you in all
essentials. ... I find in your work discussions and distinctions . . . one seeks in vain in the
works of other logicians. There is just one point where I have encountered a difficulty. ..."
Russell's paradox threw not just set theory but mathematics itself into a crisis, the third
great crisis in the history of mathematics. The first had taken place when the Pythagorean
theorem revealed to the ancient Greeks the existence of irrational numbers, those that
cannot be expressed as a ratio of two natural numbers. The second came when Newton
and Leibniz founded the infinitesimal calculus on the basis of infinitesimal numbers, which
were supposed somehow to be simultaneously nonzero and yet count for nothing. The
crises had a common cause: mathematicians found themselves confronted with a
paradoxical new kind of number. If a way could not be found to incorporate this new entity
into their thinking, they were faced with the prospect of seeing their edifice crumble.
"The sole possible foundations of arithmetic seem to vanish," Frege wrote, when
confronted with Russell's paradox.
With the third crisis, the positivists' star had risen. Mathematics itself, by its very nature as
an a priori, rationalistic science, had always been a thorn in the side of empiricists. But
now, with Cantor, mathematics had seemingly overreached itself. It had tried to fly too
high in the thin air of infinity and was in danger of crashing down on the solid earth below,
the empirical soil on which natural science is based. For mathematicians like Hilbert who
were also, in spirit, positivists, this engendered a crisis of divided loyalties. A way must be
found somehow to preserve Cantor's mathematical paradise. The answer, for Hilbert, was
to reconstruct mathematics itself along the lines of positivism. The formal proof of the
mathematician would serve as an analogue of the measuring apparatus of the empirical
scientist. Formal mathematical proofsówhich can be written down on a blackboard and
perceived with the sensesóare, no less than the instruments of the physicist, things you
can actually "get your hands on." Hilbert, then, was the Moses who would lead
mathematicians through the desert of positivism back to Cantor's paradise. He would
preserve the letter if not the spirit of Cantor's theory of infinite sets, in a manner that
satisfied the strict epistemological requirements of positivism.