Wikipedia 10K Redux by Reagle from Starling archive. Bugs abound!!!

<-- Previous | Newer --> | Current: 984168213 Gareth Owen at Fri, 09 Mar 2001 20:03:33 +0000.

Diagonal_slash_argument

A logical argument devised by [[Georg Cantor]] to demonstrate that the [[real numbers]] are not [[countably infinite]].

It proceeds as follows:
#Suppose the [[real numbers]] are countably infinite.
#We may now enumerate them as a sequence, {r1,r2,r3,...}
#We shall now construct a real number, x, by considering the nth digit of the decimal expansion of rn.  x will be between 0 and 1.
## If this nth digit of rn is 0, let the nth in digit the decimal expansion of x be 1
## Otherwise, let the nth in digit the decimal expansion of x be 0
# x is clearly a real number, but it differs in the nth decimal place from rn, so x is not in our enumeration of the real numbers.

This contradicts our supposition that the reals are countably infinite, and therefore our supposition is false.