Cantor's diagonal argument: Difference between revisions
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Cantor's diagonal method provides a convenient proof that the set <math>2^{\mathbb{N}}</math> of subsets of the [[natural number]]s (also known as its [[power set]] is not countable. More generally, it is a recurring theme in [[ | Cantor's diagonal method provides a convenient proof that the set <math>2^{\mathbb{N}}</math> of subsets of the [[natural number]]s (also known as its [[power set]] is not countable. More generally, it is a recurring theme in [[computability]] theory. | ||
==The Argument== | ==The Argument== |
Revision as of 10:35, 30 March 2007
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.
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.