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 {s}" src="https://mathworld.wolfram.com/images/equations/CantorDiagonalMethod/Inline12.gif" style="height:16px; width:14px" />. 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.

REFERENCES:

Courant, R. and Robbins, H. What Is Mathematics?: An Elementary Approach to Ideas and Methods, 2nd ed. Oxford, England: Oxford University Press, pp. 81-83, 1996.

Hoffman, P. The Man Who Loved Only Numbers: The Story of Paul Erdős and the Search for Mathematical Truth. New York: Hyperion, pp. 220-223, 1998.

Penrose, R. The Emperor's New Mind: Concerning Computers, Minds, and the Laws of Physics. Oxford, England: Oxford University Press, pp. 84-85, 1989.