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p. 857. To infinity and beyond!locked

  • Peter M. Higgins

Abstract

‘To infinity and beyond! explores the nature of infinity. Infinite sets of numbers behave differently from finite sets. Cantor proved that not all infinite sets have equally as many members, and used this to prove the existence of transcendentals. Russell's paradox shows that a set of all subsets of any countable but infinite set is uncountable. Looking at the number line shows that there are an infinite number of both rational and irrational numbers densely packed between any two numbers. However, whilst these rational numbers are countable the irrational ones are not. The Golden Ratio may allow us to represent irrational numbers through infinite continued fractions.

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