Why are the real numbers uncountable?

The real numbers are considered uncountable because they cannot be put into a one-to-one correspondence with the set of natural numbers (the counting numbers). While sets that can be paired with the natural numbers are called countably infinite (or denumerable) and have a cardinality of aleph-null (${\aleph }_0$), the real numbers possess a larger cardinality known as the continuum ($c$).123

The proof that the real numbers are uncountable was established by the mathematician Georg Cantor using a method known as the diagonal argument. This argument demonstrates that if one were to attempt to list all real numbers in a sequence, it is always possible to construct a new real number that is not included in that list by changing the digits along the diagonal of the decimal expansions.24

Because this process ensures that no complete list of real numbers can ever be created, the set of real numbers is strictly larger than the set of natural numbers, making it uncountable.12