Changes

{{Math-h}}

{{Math-h}}

'''Diagonalization''' is a technique first used by [[Cantor|Georg Cantor]], a [[Germany|German]] [[mathematician]].  He used it to show that the [[real number]]s can not be put into [[bijection|1-1 correspondence]] with the [[natural number]]s, thereby demonstrating the real numbers are not [[countable]].  This method can also be applied in other contexts.

'''Diagonalization''' is a technique first used by [[Cantor|Georg Cantor]], a [[Germany|German]] [[mathematician]].  He used it to show that the [[real number]]s can not be put into [[bijection|one-to-one correspondence]] with the [[natural number]]s, thereby demonstrating the real numbers are not [[countable]].  This method can also be applied in other contexts, to show that two sets can't have a correspondence.  For example, it can be used to show that no set can be in 1-1 correspondence with the set of all of its subsets.

==Proof of the non-countability of real numbers==

==Proof of the non-countability of real numbers==

This means that the real numbers are in 1-1 correspondence with the natural numbers if and only if the open interval <math>(0, 1)\,</math> is in correspondence.

This means that the real numbers are in 1-1 correspondence with the natural numbers if and only if the open interval <math>(0, 1)\,</math> is in correspondence.

We will now use [[proof by contradiction]] to show that the open interval has no such correspondence, and thus it, and the real line as a whole, are [[uncountable]].

We will now use [[proof by contradiction]] to show that this open interval has no such correspondence, and thus it, and the real line as a whole, are [[uncountable]].

Assume the numbers in the open interval are in a 1-1 correspondence with the natural numbers.  Then we can make an (infinite) sequential list of them, like this:

Assume the numbers in this open interval are in a 1-1 correspondence with the natural numbers.  Then we can make an (infinite) sequential list of them, like this:

<math>

<math>