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Cantor's Diagonal Argument
Also Known As Diagonalization Argument
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Cantor's diagonal argument is a proof technique Georg Cantor published in 1891 to show that the real numbers cannot be placed in one-to-one correspondence with the natural numbers: given any proposed complete listing of real numbers, a new number can always be constructed, digit by digit down the list's own diagonal, that differs from every listed number and so cannot appear on it. Cantor later generalised the same technique to prove Cantor's theorem, that a set's power set is always strictly larger than the set itself.
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1. Wikipedia: Cantor's Diagonal Argument
Wikimedia FoundationIntroductionQuote, Introduction
Cantor's diagonal argument (among various similar names) 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.
1. Wikipedia: Cantor's Diagonal Argument
Wikimedia FoundationLead sectionQuote, Lead section
Georg Cantor published this proof in 1891.
1. Wikipedia: Cantor's Diagonal Argument
Wikimedia FoundationIn Category: Concepts, lead paragraph, opening definitionQuote, In Category: Concepts, lead paragraph, opening definition
Cantor's diagonal argument (among various similar names) 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
1. Wikipedia: Cantor's Diagonal Argument
Wikimedia FoundationIn Branch: Logic and Foundations, lead paragraph, applications passageQuote, In Branch: Logic and Foundations, lead paragraph, applications passage
it demonstrates a general technique that has since been used in a wide range of proofs, including the first of Godel's incompleteness theorems and Turing's answer to the Entscheidungsproblem.
1. Wikipedia: Cantor's Diagonal Argument
Wikimedia FoundationAssociated With: Georg Cantor, lead paragraph, publication dateQuote, Associated With: Georg Cantor, lead paragraph, publication date
Georg Cantor published this proof in 1891
1. Wikipedia: Cantor's Diagonal Argument
Wikimedia FoundationAssociated With: Cantor's Theorem, Cantor's theorem sectionQuote, Associated With: Cantor's Theorem, Cantor's theorem section
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, cannot be in bijection with S itself.
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