Bill's post on how to derive the non-finiteness of the primes from Van der Waerden's theorem reminds me of a nice proof using Kolmogorov complexity. A quick primer: Fixed some universal programming language. Let C(x), the Kolmogorov complexity of x, be the length of the smallest program that outputs x. One can show by a simple counting argument for every n there is an x such that C(x) ≥ n.