Cantor's theorem: Difference between revisions

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In mathematical [[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>) 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]].

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> elements has a total of <math>2^n</math> subsets, 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>.

Much more significant is Cantor's discovery of an argument that is applicable to any set, and shows that the theorem holds for [[infinite set|infinite]] sets also. As a consequence, the cardinality of the [[real number]]s, which is the same as that of the power set of the [[integer]]s, is strictly larger than the cardinality of the integers; see [[Cardinality of the continuum]] for details.

The theorem is named for German [[mathematician]] [[Georg Cantor]], who first stated and proved it at the end of the 19th century. Cantor's theorem had immediate and important consequences for the [[philosophy of mathematics]]. For instance, by iteratively taking the power set of an infinite set and applying Cantor's theorem, we obtain an endless hierarchy of infinite cardinals, each strictly larger than the one before it. Consequently, the theorem implies that there is no largest cardinal number (colloquially, "there's no largest infinity").