Cantor's diagonal argument

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Revision as of 10:04, 30 March 2007 by imported>Greg Woodhouse (ref. to halting problem)
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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 computability theory, where perhaps its most well known application is the negative solution halting problem.

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.