A set is said to be uncountable if there is no injection . Assuming the Axiom of choice, every set that is not uncountable is either finite or countably infinite. The most common example of an uncountable set is the set of real numbers .
Proof that is uncountable
We give an indirect proof here. This is one of the most famous indirect proofs and was first given by Georg Cantor.
Suppose that the set is countable. Let be any enumeration of the elements of (in other words, take an injection , and denote ).
Consider the decimal expansion of each , say for all . Now construct a real number , by choosing the digit so that it differs from by at least 3 and so that is never equal to 9 or 0. It follows that differs from by at least , so for every . Thus, . However, is clearly a real number between 0 and 1, a contradiction. Thus our assumption that is countable must be false, and since we have that is uncountable.
An alternative proof uses Cantor's Theorem, which says that for all sets , we have , where is the power set of . First, we note that the Cantor set has cardinality , and since , there is an injection and thus , so is uncountable. In fact, it turns out that .
This article is a stub. Help us out by expanding it.