Cantor's theorem: Difference between revisions

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In elementarymathematical [[set theory]], '''Cantor's theorem''' is a fundamental result which states that, for any [[Set (mathematics)|set]] <math>A</math>, the set of all [[subset]]s of <math>A</math> (the [[power set]] of <math>A</math>, denoted by <math>\mathcal{P}(A)</math>) has a strictly greater [[cardinality]] than <math>A</math> itself. For [[finite set]]s, Cantor's theorem can be seen to be true by simple [[enumeration]] of the number of subsets. Counting the [[empty set]] as a subset, a set with <math>n</math> members has a total of <math>2^n</math> subsets, so that if <math>\operatorname{card}(A) = n,</math> then <math>\operatorname{card}(\mathcal{P}(A)) = 2^n</math>, and the theorem holds because <math>2^n > n</math> for all [[non-negative integers]].

Much more significant is Cantor's discovery of an argument that is applicable to any set, which showed that the theorem holds for [[infinite set|infinite]] sets, countable or uncountable, as well as finite ones. As a particularly important consequence, the power set of the set of [[natural number]]s, a [[countably infinite]] set with cardinality {{math|1=ℵ₀ = card('''N''')}}, is [[uncountable set|uncountably infinite]] and has the same size as the set of [[real number]]s, a cardinality larger than that of the set of natural numbers that is often referred to as the [[cardinality of the continuum]]: {{math|1=𝔠 = card('''R''') = card(𝒫('''N'''))}}. The relationship between these cardinal numbers is often expressed symbolically by the equality and inequality <math>\mathfrak{c} = 2^{\aleph_0} > \aleph_0</math>.