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). It proceeds by first considering a countably infinite list of elements from a set , each of which is an infinite set (in the case of the Reals, the decimal expansion of each Real). A new member of is then created by arranging its th term to differ from the th term of the th member of . This shows that is not Countable, since any attempt to put it in one-to-one correspondence with the integers will fail to include some elements of .

**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.

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

© 1996-9

1999-05-26