Editing Cantor's diagonal argument (section)

===General sets===
[[File:Diagonal argument powerset svg.svg|thumb|250px|Illustration of the generalized diagonal argument: The set <math>T = \{n \in \mathbb{N}: n \not\in f(n)\}</math> at the bottom cannot occur anywhere in the [[Range of a function|range]] of <math>f:\mathbb{N}\to\mathcal{P}(\mathbb{N})</math>. The example mapping ''f'' happens to correspond to the example enumeration ''s'' in the picture [[#Lead|above]].]]
A generalized form of the diagonal argument was used by Cantor to prove [[Cantor's theorem]]: for every [[Set (mathematics)|set]] ''S'', the [[power set]] of ''S''—that is, the set of all [[subset]]s of ''S'' (here written as '''''P'''''(''S''))—cannot be in [[bijection]] with ''S'' itself. This proof proceeds as follows:

Let ''f'' be any [[Function (mathematics)|function]] from ''S'' to '''''P'''''(''S''). It suffices to prove that ''f'' cannot be [[surjective]]. This means that some member ''T'' of '''''P'''''(''S''), i.e. some subset of ''S'', is not in the [[Image (mathematics)|image]] of ''f''. As a candidate consider the set
: <math>T = \{ s \in S : s \notin f(s) \}.</math>

For every ''s'' in ''S'', either ''s'' is in ''T'' or not. If ''s'' is in ''T'', then by definition of ''T'', ''s'' is not in ''f''(''s''), so ''T'' is not equal to ''f''(''s''). On the other hand, if ''s'' is not in ''T'', then by definition of ''T'', ''s'' is in ''f''(''s''), so again ''T'' is not equal to ''f''(''s''); see picture.

For a more complete account of this proof, see [[Cantor's theorem]].