Uncountable
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A set is said to be uncountable if there is no injection . A well-known 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 given by George Cantor.
Suppose that the set is countable. Let be any enumeration of the elements of . Consider the binary expansion of each . For diedic rationals, take only the expansion terminating in an infinite chain of zeros, so that we have a unique binary expansion for each element of . Say for all . Now construct such that for all . So is different from for all . So . But , a contradiction. So is uncountable.