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?



Logic comment in Mumford's Red Book
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.


