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The example mapping f happens to correspond to the example enumeration s in the picture above. 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—that is, the set of all subsets of S (here written as P(S))—cannot be in bijection with S itself. This proof proceeds as ...
The answer is yes, because for every natural number n there is a square number n 2, and likewise the other way around. The answer is no, because the squares are a proper subset of the naturals: every square is a natural number but there are natural numbers, like 2, which are not squares of natural numbers.
Cantor's theorem and its proof are closely related to two paradoxes of set theory. Cantor's paradox is the name given to a contradiction following from Cantor's theorem together with the assumption that there is a set containing all sets, the universal set. In order to distinguish this paradox from the next one discussed below, it is important ...
Cantor's diagonal argument has often replaced his 1874 construction in expositions of his proof. The diagonal argument is constructive and produces a more efficient computer program than his 1874 construction. Using it, a computer program has been written that computes the digits of a transcendental number in polynomial time.
This is known as Cantor's theorem. The argument below is a modern version of Cantor's argument that uses power sets (for his original argument, see Cantor's diagonal argument). By presenting a modern argument, it is possible to see which assumptions of axiomatic set theory are used.
Cantor's diagonal argument does not also work for fractional rational numbers because the "anti-diagonal real number" is indeed a fractional irrational number --- hence, the presence of the prefix fractional expansion point is not a consequence nor a valid justification for the argument that Cantor's diagonal argument does not work on integers.
Cantor gave two proofs that the cardinality of the set of integers is strictly smaller than that of the set of real numbers (see Cantor's first uncountability proof and Cantor's diagonal argument). His proofs, however, give no indication of the extent to which the cardinality of the integers is less than that of the real numbers.
Later, in 1891, Cantor used his more familiar diagonal argument to prove the same result. [17] While Cantor's result is often quoted as being purely existential and thus unusable for constructing a single transcendental number, [18] [19] the proofs in both the aforementioned papers give methods to construct transcendental numbers. [20]