Uncountable set
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In mathematics, an uncountable set is a set which is not countable. Here, "countable" means countably infinite or finite, so by definition, all uncountable sets are infinite. Explicitly, a set X is uncountable if and only if there does not exist a surjective function from the natural numbers N to X.
Not all uncountable sets have the same size; the sizes of infinite sets are analyzed with the theory of cardinal numbers. Formally, an uncountable set is defined as one whose cardinality is strictly greater than <math>\aleph_0<math> (aleph-null), the cardinality of the natural numbers.
The best known example of an uncountable set is the set R of all real numbers; Cantor's diagonal argument shows that this set is uncountable. The diagonalization proof technique can also be used to show that several other sets are uncountable as well, for instance the set of all infinite sequences of natural numbers (and even the set of all infinite sequences consisting only of zeros and ones) and the set of all subsets of natural numbers. The cardinality of R is often called the cardinality of the continuum and denoted by c or <math>\beth_1<math> (beth-one).
The Cantor set is an uncountable subset of R. The Cantor set is a fractal and has Hausdorff dimension greater than zero but less than one. (R has dimension one.) This is an example of the following fact: any subset of R of Hausdorff dimension strictly greater than zero must be uncountable.
Another example of a uncountable set is the set of all functions from R to R. It is not hard to believe that this set is even "more uncountable" than R. The cardinality of this set is <math>\beth_2<math> (beth-two) and is, indeed, larger than <math>\beth_1<math>.
A much more abstract example of an uncountable set is the set of all countably infinite ordinal numbers, denoted Ω. The cardinality of Ω is denoted <math>\aleph_1<math> (aleph-one). It can be shown that <math>\aleph_1<math> is the smallest uncountable cardinal number. One might naturally wonder whether <math>\beth_1<math>, the cardinality of the reals, is equal to <math>\aleph_1<math> or if it is strictly larger. The statement that <math>\aleph_1 = \beth_1<math> is called the continuum hypothesis. This hypothesis is now known to be independent from the ordinary axioms of set theory (cf. Zermelo-Frankel axioms). Which is to say, that one can either assume the continuum hypothesis is true, or assume that is false without running into any contradictions.