Ultimate logic: To infinity and beyond
WHEN David Hilbert left the podium at the Sorbonne in Paris, France, on 8 August 1900, few of the assembled delegates seemed overly impressed. According to one contemporary report, the discussion following his address to the second International Congress of Mathematicians was “rather desultory”. Passions seem to have been more inflamed by a subsequent debate on whether Esperanto should be adopted as mathematics’ working language.
Yet Hilbert’s address set the mathematical agenda for the 20th century. It crystallised into a list of 23 crucial unanswered questions, including how to pack spheres to make best use of the available space, and whether the Riemann hypothesis, which concerns how the prime numbers are distributed, is true.
Today many of these problems have been resolved, sphere-packing among them. Others, such as the Riemann hypothesis, have seen little or no progress. But the first item on Hilbert’s list stands out for the sheer oddness of the answer supplied by generations of mathematicians since: that mathematics is simply not equipped to provide an answer.
This curiously intractable riddle is known as the continuum hypothesis, and it concerns that most enigmatic quantity, infinity. Now, 140 years after the problem was formulated, a respected US mathematician believes he has cracked it. What’s more, he claims to have arrived at the solution not by using mathematics as we know it, but by building a new, radically stronger logical structure: a structure he dubs “ultimate L”.
Because there is more to life than politics, religion and the economy. The following passage alone should give you some sleepless nights:
The axiom of choice states that if you have a collection of sets, you can always form a new set by choosing one object from each of them. That sounds anodyne, but it comes with a sting: you can dream up some twisted initial sets that produce even stranger sets when you choose one element from each. The Polish mathematicians Stefan Banach and Alfred Tarski soon showed how the axiom could be used to divide the set of points defining a spherical ball into six subsets which could then be slid around to produce two balls of the same size as the original.
Subset clones? Something from nothing? AHHHHHHHHHHHHH!!!!! (Don’t despair - Woodin Cardinals put these monstrosities in their place).
I have a sudden urge to find my unread copy of Gödel, Escher, Bach: An Eternal Golden Braid, which I bought in an (unsuccessful) attempt to look cool and intellectual on airplanes.