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 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
In Branch
Source Cantor's Theorem (Wikipedia)
Named After
Derived from the theorem's own name (unambiguous possessive-token match to exactly one live mathematician entity, w-bfill-g5-0924 browse backfill)
Proved By
Source Cantor's Theorem (Wikipedia)
Sources
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
View the Source 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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