Probability paradox

A result that amazed me when I first heard of it is that Gaussian measure on infinite dimensional Hilbert spaces does not exist.

At first sight it seems obvious that it exists: on any finite dimensional quotient one can define a Gaussian probability measure, and these are all “compatible”. So their “inverse limit” should give a Gaussian probability measure on the original infinite dimensional Hilbert space.

In fact this cannot work: the ball of radius R in 1 dimensional space has Gaussian volume V<1, so the ball of radius R in n dimensions has Gaussian measure at most which tends to 0 as n tends to infinity. So the ball of radius R in infinite dimensions has Gaussian measure 0 for any R. As Hilbert space is the union of a countable number of such balls, it has Gaussian measure 0, contradicting the fact that Gaussian measure gives it measure 1.

In fact it is possible to construct Gaussian measure on a Hilbert space H, but it has support larger than H. More precisely, if S is a Hilbert-Schmidt operator from H to K, then although Gaussian measure is not well defined, its image under S is a well defined probability measure on K (Sazonov’s theorem).

The non-existence of Gaussian measure on infinite dimensional Hilbert spaces is one of the things that makes quantum (or rather Euclidean) field theory hard: roughly speaking the functions one wants to integrate are only defined on H and not on the larger space K.

Uselessness of set theory?

Mathematics is supposedly based on set theory. But if one looks closely, it is amazing (to me) just how little set theory is really used. Almost all “ordinary” mathematics seems to use no sets larger than the continuum. In other words most of mathematics can be carried out in second order arithmetic: roughly speaking this means that one can use the integers and subsets of the integers but nothing more complicated. Although third order and higher order constructions occasionally occur, it usually seems easy to replace them by second order constructions. (By ordinary math I mean things like the Atiyah-Singer index theorem, Fermat’s last theory, the Weil conjectures, and so on; more or less everything that is not some sort of set theory or logic.)

In fact one seems to need a lot less than this if one is willing to work harder. I suspect that a lot of stuff can be encoded in Peano arithmetic if one is willing to work hard (for example, encoding constructible reals in terms of integers is tiresome but possible). The few examples known of theorems that cannot be proved using Peano arithmetic (as in the Paris-Harrington theorem) tend to have extremely rapidly growing functions appearing, and I would guess that proofs without such large functions somewhere can usually be encoded in Peano arithmetic. One can probably go a lot lower: Peano arithmetic has much weaker fragments, such a primitive recursive arithmetic. In practice it seems rare to have functions that require more than a finite tower of exponentials to describe, which presumably corresponds to something even weaker than primitive recursive arithmetic (does anyone know what this is called?).

Harvey Friedman has a program called “reverse mathematics” to identify what axioms are really needed to prove various results, but this seems to concentrate on various fragments of second order arithmetic. He has found many examples of results about the integers, often with a Ramsey-theoretic flavor, that need reasonably strong subsystems of second order arithmetic to prove.

It is easy to find mathematical results whose proofs need much stronger systems using Godel’s theorem: for example, the consistency of second order arithmetic is a perfectly good statement about integers that is (presumably…) true, but cannot be proved in second order arithmetic, though it can easily be proved even in weak set theories. I do not know of any mathematical theorems about the integers that cannot be proved in second order arithmetic that are not related to consistency results.

So my question is the following: why do we use so little set theory in ordinary mathematics? Is it because almost all interesting mathematical results only need tiny fragments of Peano arithmetic to use, or is is because we are too stupid to make proper use of the vastly more powerful axioms in set theory?