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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.
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 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 ...
Diagonal argument can refer to: Diagonal argument (proof technique), proof techniques used in mathematics. A diagonal argument, in mathematics, is a technique employed in the proofs of the following theorems: Cantor's diagonal argument (the earliest) Cantor's theorem; Russell's paradox; Diagonal lemma. Gödel's first incompleteness theorem
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 ...
The evidence strongly suggests that Cantor was quite confident in the result itself and that his comment to Dedekind refers instead to his then-still-lingering concerns about the validity of his proof of it. [1] Nevertheless, Cantor's remark would also serve nicely to express the surprise that so many mathematicians after him have experienced ...
Cantor's back-and-forth method; Cantor's diagonal argument; Cantor's first uncountability proof; Cantor's paradox; Cantor's theorem; Cantor–Bernstein–Schroeder theorem; Cardinal number. Aleph number; Beth number; Hartogs number; Cardinality; Cartesian product; Class (set theory) Complement (set theory) Complete Boolean algebra; Continuum ...
This theorem is proved using Cantor's first uncountability proof, which differs from the more familiar proof using his diagonal argument. The title of the article, " On a Property of the Collection of All Real Algebraic Numbers " ("Ueber eine Eigenschaft des Inbegriffes aller reellen algebraischen Zahlen"), refers to its first theorem: the set ...