The Cantor diagonal method, also called the Cantor diagonal argument or Cantor's diagonal slash, is a clever technique used by Georg Cantor to show that the integers and reals cannot be put into a one-to-one correspondence (i.e., the uncountably infinite set of real numbers is "larger" than the countably infinite set of integers). However, Cantor's diagonal method is completely general and applies to any set as described below.
Given any set , consider the power set consisting of all subsets of . Cantor's diagonal method can be used to show that is larger than , i.e., there exists an injection but no bijection from to . Finding an injection is trivial, as can be seen by considering the function from to which maps an element of to the singleton set . Suppose there exists a bijection from to and consider the subset of consisting of the elements of such that does not contain . Since is a bijection, there must exist an element of such that . But by the definition of , the set contains if and only if does not contain . This yields a contradiction, so there cannot exist a bijection from to .
Cantor's diagonal method applies to any set , finite or infinite. If is a finite set of cardinality , then has cardinality , which is larger than . If is an infinite set, then is a bigger infinite set. In particular, the cardinality of the real numbers , which can be shown to be isomorphic to , where is the set of natural numbers, is larger than the cardinality of . By applying this argument infinitely many times to the same infinite set, it is possible to obtain an infinite hierarchy of infinite cardinal numbers.