Liouville Numbers

Abstract

Cantor’s theorem, Baire’s theorem, and Borel’s theorem are existence theorems. If one can show that the set of numbers in an interval that lack a certain property is either countable, or a nullset, or a set of first category, then it follows that there exist points of the interval that have the property in question, in fact, most points of the interval (in the sense of cardinal number, or measure, or category, respectively) have the property. As a first illustration of this method let us consider the existence of transcendental numbers.