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Assume (for the sake of argument) that the interval [0,1] is countably infinite.
A generalized form of the diagonalargument 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.
Analogues of the diagonalargument are widely used in mathematics to prove the existence or nonexistence of certain objects.
(It is also called the diagonalizationargument or the diagonal slash argument.) It does this by showing that the interval (0,1), that is, the set of real numbers larger than 0 and smaller than 1, is not countably infinite.
The diagonalargument is an example of reductio ad absurdum because it proves a certain proposition (the interval (0,1) is not countably infinite) by showing that the assumption of its negation leads to a contradiction.
A generalized form of the diagonalargument was used by Cantor to show that 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.
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