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Cantor's diagonal argument (among various similar names [note 1]) is a mathematical proof that there are infinite sets which cannot be put into one-to-one correspondence with the infinite set of natural numbers – informally, that there are sets which in some sense contain more elements than there are positive integers.
Much more significant is Cantor's discovery of an argument that is applicable to any set, and shows that the theorem holds for infinite sets also. As a consequence, the cardinality of the real numbers , which is the same as that of the power set of the integers , is strictly larger than the cardinality of the integers; see Cardinality of the ...
Cantor's paradox: The set of all sets would have its own power set as a subset, therefore its cardinality would be at least as great as that of its power set. But Cantor's theorem proves that power sets are strictly greater than the sets they are constructed from. Consequently, the set of all sets would contain a subset greater than itself.
In the same year the French mathematician Jules Richard used a variant of Cantor's diagonal method to obtain another contradiction in naive set theory. Consider the set A of all finite agglomerations of words. The set E of all finite definitions of real numbers is a subset of A.
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.
The best known example of an uncountable set is the set of all real numbers; Cantor's diagonal argument shows that this set is uncountable. The diagonalization proof technique can also be used to show that several other sets are uncountable, such as the set of all infinite sequences of natural numbers (see: (sequence A102288 in the OEIS)), and the set of all subsets of the set ...
Cantor recovered soon thereafter, and subsequently made further important contributions, including his diagonal argument and theorem. However, he never again attained the high level of his remarkable papers of 1874–84, even after Kronecker's death on 29 December 1891. [ 20 ]
However, this book is the first to focus on mathematical crankery as a whole. [1] The book consists of 57 essays, [2] loosely organized by the most common topics in mathematics for cranks to focus their attention on. [1] The "top ten" of these topics, as listed by reviewer Ian Stewart, are, in order: squaring the circle, angle trisection,