Heller M, Woodin W.H. (eds.) Infinity. New research frontiers (CUP, 2011)(ISBN 1107003873)(O)(327s)_MAml_

a success of set theory 69A fundamental concept of geometry is the notion of boundary. In antiquity, anargument given in favor of an infinite universe was the following: If the universe isfinite, what lies beyond its boundary? If the earth is flat, what is its boundary and whatcould be beyond it? Against this argument stood the Zeno paradox, which could beused to explain that the boundary could never be reached, and the examples of thecircle or the sphere, which are finite but without boundary.Today, mathematical models can capture all such possibilities by means of simpleexamples. A disk with hyperbolic metric has its circumference as its boundary, butany curve reaching the boundary must have infinite length, making the boundary“unreachable.” This is an example of a manifold with a complete metric. If instead weadopt the usual Euclidean metric, we can reach the boundary following a path of finitelength, so the metric here is not complete. Geometers have studied ways by whichone may enlarge a space with a metric that is not complete, so, to extend the space andthe metric to a new space with a complete metric. This leads to new objects that captureall of the previous space but also have much better geometric properties, because pathswill never stop abruptly at the boundary and can always be extended to infinite length.Also, geometry in infinite dimensions has become a basic tool in mathematics, withthe theory of Hilbert spaces extending Euclidean geometry in the plane and space.The quest for understanding the shape of our universe has gone today way beyond thesimplistic models of the past, adding a time dimension to the usual three-dimensionalspace, adding curvature to space as in general relativity, and even adding an extra six orseven dimensions to unify microcosm and macrocosm as in the current models of theuniverse proposed by string theorists. Certainly, this challenge to unify the universe ofthe galaxies with the subatomic universe is one of the most exciting aspects of moderntheoretical physics. If the universe is finite, as seems to be the consensus at the presentmoment, what will be its geometry?2.9 A Success of Set TheoryIt would be wrong to consider set theory as an esoteric small branch of mathematics,an end unto itself. It has produced a number of significant results in pure mathematics,all the way to elementary number-theoretic statements. Here is my favorite example ofthis.A diophantine equation is, in its simplest version,f (x 1 ,x 2 ,...,x n ) = 0,where f is a polynomial with integer coefficients, to be solved in integers x 1 , x 2 , ...,x n . A related problem, namely, the solution in rational numbers, is a special case ofthe above when the polynomial f is homogeneous, that is, composed of monomials allwith the same total degree d (excluding now the trivial zero solution).Because zero is the only integer divisible by every natural integer, the first stepin solving a diophantine equation is asking the solvability of a divisibility problem,namely, whether for any natural integer N we can find integers x 1 , ..., x n (not all 0 if fis homogeneous) such that f (x 1 ,...,x n ) is divisible by N. It is quite easy to show thatwe need to prove such a thing just when N is a prime power p n , for all primes p and all