Wikipedia 10K Redux

Reconstructed by Reagle from Starling archive; see blog post for context.

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. Our real number x will be between 0 and 1.

    • ::If the 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.

    The diagonal slash argument is an example of Reductio ad absurdum.