General Sets
A generalized form of the diagonal argument was used by Cantor to prove Cantor's theorem: for every set S the power set of S, i.e., the set of all subsets of S (here written as P(S)), is larger than S itself. This proof proceeds as follows:
Let f be any function from S to P(S). It suffices to prove f cannot be surjective. That means that some member T of P(S), i.e., some subset of S, is not in the image of f. As a candidate consider the set:
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). For a more complete account of this proof, see Cantor's theorem.
Read more about this topic: Cantor's Diagonal Argument
Famous quotes containing the words general and/or sets:
“It is a maxim among these lawyers, that whatever hath been done before, may legally be done again: and therefore they take special care to record all the decisions formerly made against common justice and the general reason of mankind.”
—Jonathan Swift (1667–1745)
“We are amphibious creatures, weaponed for two elements, having two sets of faculties, the particular and the catholic.”
—Ralph Waldo Emerson (1803–1882)