Advanced Mathematics

2.5Uncountable Sets

Purpose of Section

We now present Cantor's seminal result that the cardinality of real numbers is uncountable. We then continue with the equally exciting result that the cardinality of the two‐dimensional plane is the same as the cardinality of the one‐dimensional real line.

2.5.1 Introduction

Thus far, the only infinity considered has been countable infinity, which is the cardinality of the natural numbers or any set of many sets that can be written as a sequence. This begs the question are all infinite sets equivalent to the natural numbers? In 1874, the German mathematician Georg Cantor proved that there are in fact sets with even larger cardinalities than of the natural numbers, which led to the development of modern set theory.

At the time Cantor believed, as did all mathematicians, that infinity was infinity. Thinking along those lines, Cantor tried to prove that the real numbers have the same cardinality as the natural numbers, but failed and his failure was one of the greatest failures in the history of mathematics since in the process he proved that the real numbers have a larger cardinality than that of the natural numbers.

Definition

An infinite set with cardinality larger than that of the natural numbers is said to have uncountable cardinality, or simply an uncountable set.

The above discussion leads us to Cantor's famous diagonalization theorem.

Theorem 1 Cantor's Diagonalization Theorem

The set of real numbers in the open ...