Cantor's theorem is a fundamental result of set theory stating that for any set A, the power set of A, the set of all its subsets, has a strictly greater cardinality than A itself. More concisely, every set is smaller than its power set. This description is adapted from Wikipedia contributors under CC BY-SA 4.0; changes were made. https://creativecommons.org/licenses/by-sa/4.0/
Facts
StatementFor any set, the collection of all of its subsets always has strictly more elements, in the cardinality sense, than the original set itself. 1 Proof Method Classification
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Associated With
Source Wikipedia: Cantor's Diagonal Argument
Sets, Concepts The theorem compares the cardinality of a set to its own power set, so its subject is the set concept itself.
Additional Source Cantor's Theorem (Wikipedia)lead
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Source Cantor's Theorem (Wikipedia)
Named After
Principal Result Of
Source Wikipedia: Georg Cantor
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Source Cantor's Theorem (Wikipedia)
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1. Cantor's Theorem (Wikipedia)
Wikimedia Foundationlead
Every set is smaller than its power set
opening sentence
for any set A, the set of all subsets of A, known as the power set of A, has a strictly greater cardinality than A itself
proof and attribution section
the diagonal argument for the uncountability of the reals also first appears
In Group: Set Theory, Lead section
In mathematical set theory, Cantor's theorem is a fundamental result which
View the Source 2. ProofWiki: Cantor's Theorem
ProofWikiProof sectionQuote, Proof section
Aiming for a contradiction, suppose $S$ is a set with a surjection $f: S \to \powerset S$.
Wikipedia: Cantor's Diagonal Argument
Wikimedia FoundationAssociated With: Cantor's Diagonal Argument, General setsQuote, Associated With: Cantor's Diagonal Argument, General sets
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.
Wikipedia: Georg Cantor
WikipediaPrincipal Result Of: Georg Cantor, Article textQuote, Principal Result Of: Georg Cantor, Article text
He later proved that the size of the power set of A is strictly larger than the size of A, even when A is an infinite set; this result soon became known as Cantor's theorem.
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