Revision as of 10:34, 30 March 2007 by imported>Greg Woodhouse
Cantor's diagonal method provides a convenient proof that the set of subsets of the natural numbers (also known as its power set is not countable. More generally, it is a recurring theme in combutability theory.
The Argument
To any set we may associate a function by setting if and , otherwise. Conversely, every such function defines a subset.
If power set is countable, there is a bijective map , that allows us to assign an index to every subset S. Assuming this has been done, we proceed to construct a function such that the corresponding set, cannot be in the range of .
For each , either or , and so we may simply such that .
It follows that for any , and it must therefore correspond to a set not in the range of . This contradiction shows that cannot be countable.