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

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. In 1874, two years before the publication of his famous diagonalization argument, Georg Cantor’s first proof of the uncountability of the real numbers appeared in print [1]. Surprisingly, a small twist on Cantor’s line of reasoning shows that the Golden Ratio is irrational, as we shall demonstrate herewith. En route, we will make use of another classic 19th century theorem by another Georg, this time Georg Pick. Pick’s Theorem [2] provides a simple formula for the number of lattice points enclosed within a simply connected polygonal region in the plane with lattice point vertices. We begin by recapitulating Cantor’s 1874 proof. To show that the real numbers are uncountable, we must show that given any countable sequence of distinct real numbers, there exists another real number not in the sequence. Like the diagonalization argument, we will do so by providing an explicit alogrithm which produces such a number; unlike the diagonalization argument, we will employ not decimal expansions but order properties of the real numbers. Let (an) be a countable sequence of distinct real numbers. Suppose that there are two distinct terms aj and ak such that no term a` lies strictly between aj and ak—in other words, suppose that {an} does not possess the Intermediate Value Property. Let L be any real number strictly between aj and ak, for example aj + ak 2 . Then L is not in the sequence (an). Now suppose that (an) does have the Intermediate Value Property. Cantor recursively constructs two new sequences (bn) and (cn) as follows. Let b1 = a1, and let c1 = a2. Let bk+1 be the first term in (an) that lies strictly between bk and ck. Let ck+1 be the first term in (an) that lies strictly between bk+1 and ck.