In Mumford's "The red book of varieties and schemes" one of the examples (G on pg 74) is the space Spec $(\prod_{i=1}^\infty k)$, where $k$ is a field. He mentions that "Logicians assure us that we can prove more theorems if we use these outrageous spaces".
Are the any examples of theorems proved using such spaces, or any references to logicians saying such a thing?
At (about) the time Mumford was giving his lectures at Harvard, Ax was lecturing on his work with Kochen in which they proved a conjecture of Artin for almost all p by using ultrafilters. This is clearly what Mumford was thinking of. The reference for the Ax-Kochen work is:
MR0184930 Ax, James; Kochen, Simon Diophantine problems over local fields. I. Amer. J. Math. 87 1965 605--630. Ib. 87 1965 631--648.
These things have to do with ultrafilters. This is not my field, but http://www.math.sc.edu/~nyikos/rings1.ps seems like an okay introduction to the area. I think that if k is a ring and X is an infinite set, Spec(k^X) can be identified with the set of ultrafilters on X. The theorem that every ideal is contained in a maximal ideal implies the existence of non-principal ultrafilters (which correspond to maximal ideals of k^X other than the obvious ones), which logicians use to make constructions of non-standard models of the reals and the like.
Like Alison said, one can identify $Spec(k^{\mathbb{N}})$ with the set of ultrafilters on $\mathbb{N}$. There is a canonical topology on this set, which makes it into the Stone-Cech compactification of $\mathbb{N}$, $S \mathbb{N}$: one takes as a basis the sets $U_A = ${$ F \in S \mathbb{N} : A \in F $} , where $A \subset \mathbb{N}$.
$S \mathbb{N}$ is universal among compactifications of $ \mathbb{N}$, in the sense that every map from $\mathbb{N}$ to compact $X$ extends uniquely to $S\mathbb{N}$. It's not hard to see that $S \mathbb{N}$ is homeomorphic to $(Spec(k^{\mathbb{N}}), zariski)$.
I think that what Mumford is pointing at are Stone-Cech compactifications in general rather than $Spec(k^{\mathbb{N}})$ in particular. There's quite a good brief definition and explanation of them in Steen & Seebach's `Counterexamples in Topology'; you could also check out the references on the Wikipedia page.
If you take a product of finite fields of infinitely many characteristics and divide by a maximal ideal, the result is called a pseudo-finite field. This has characteristic zero and a commutative Galois group: Z-hat.
This is decades later than Mumford's book, but Tomasic constructed a Weil cohomology theory using pseudofinite fields as coefficients:
http://www.maths.qmul.ac.uk/~ivan/psf-weil-coh.ps
The construction still uses etale cohomology, but only with finite coefficients. But I guess the verification that it satisfies the Weil axioms still makes use of the ell-adic theory.
Non-triviality of an infinite product is a reformulation of the Axiom of Choice. So possibly he is referring to the fact that (logicians say that) the Axiom of Choice is independent, and has some consequences that cannot be proved without it.
