Theorems
Cantor's Theorem
KAN-tor
Also Known As Cantor's Power Set Theorem
Logic and Foundations
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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.
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 Cross-Tradition Connections
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1. Cantor's Theorem (Wikipedia)
Wikimedia FoundationleadQuote, lead
Every set is smaller than its power set
1. Cantor's Theorem (Wikipedia)
Wikimedia Foundationopening sentenceQuote, 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
1. Cantor's Theorem (Wikipedia)
Wikimedia Foundationproof and attribution sectionQuote, proof and attribution section
the diagonal argument for the uncountability of the reals also first appears
Wikipedia: Cantor's Diagonal Argument
Wikimedia FoundationAssociated With: Cantor's Diagonal Argument, Cantor's theorem sectionQuote, Associated With: Cantor's Diagonal Argument, 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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