Metaphors in teaching infinity: Limits, cardinalities, and nonstandard models

Abstract

Mathematical conception of infinite quantities forms a cornerstone of many disciplines of modern mathematics --- from differential calculus to set theory. In fact, it could be argued that the most significant revolutions in mathematics in the modern period were always triggered by a development in our understanding of infinity. From the pedagogical point of view, the students' comprehension of the concept of infinity is a competence of interdisciplinary value, helping them to grasp the reasons why and how individual disciplines of modern mathematics were built up. In this paper we present a number of illustrations, examples, and allegories that illuminate and clarify different aspects of infinity. The author has been using and gradually developing these metaphors for infinity in his undergraduate and graduate courses and popular lectures on mathematical logic, set theory, calculus, and nonstandard analysis during the last decade. The aim of this paper is to share the accumulated didactic know-how.