On Cantor's First Uncountability Proof, Pick's Theorem, and the Irrationality of the Golden Ratio

Abstract

In Cantor's original proof of the uncountability of the reals (not the diagonalization argument), he constructs, given any countable sequence of real numbers, a real number not in the sequence. When we apply this argument to a certain standard enumeration of the rationals, the real number we produce will necessarily be irrational. Using some planar geometry, including Pick's theorem on the number of lattice points enclosed within certain polygonal regions, we show that this number is the reciprocal of the golden ratio, whence follows the well-known fact that the golden ratio is irrational.