Limited enumeration of real numbers by lists of bits illustrates Cantor's diagonalization argument. The number formed from complements of bits on the diagonal is not included because of the way it is constructed. All permutations of the enumerations must miss at least the number corresponding to the inverted digits on the diagonal.

The real numbers in the unit interval are usually regarded as a model of the continuum. Cantor's continuum hypothesis states that there are no sets with a cardinality between the cardinality of natural numbers and the cardinality of the continuum.