Suppose were countable, and list the reals as . Write each in decimal form:

Define by if , and if . Then has only one decimal representation and differs from each at the th decimal place, so no list exhausts . This is known as Cantor’s diagonal argument.

The argument also shows is uncountable, and hence any interval in is uncountable.

There are uncountably many transcendental numbers: if there were only countably many, then together with the countable set of algebraic numbers they would express as a countable union of countable sets, contradicting the uncountability of .