Uncountable set

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In mathematics, an uncountable set is an infinite set that contains too many elements to be countable. The uncountability of a set is closely related to its cardinal number: a set is uncountable if its cardinal number is larger than that of the set of all natural numbers.

Contents

Characterizations

There are many equivalent characterizations of uncountability. A set X is uncountable if and only if any of the following conditions holds:

  • There is no injective function from X to the set of natural numbers.
  • X is nonempty and any ω-sequence of elements of X fails to include at least one element of X. That is, X is nonempty and there is no surjective function from the natural numbers to X.
  • The cardinality of X is neither finite nor equal to \aleph_0 (aleph-null, the cardinality of the natural numbers).
  • The set X has cardinality strictly greater than \aleph_0.

The first three of these characterizations can be proved equivalent in Zermelo–Fraenkel set theory without the axiom of choice, but the equivalence of the third and fourth cannot be proved without additional choice principles.

Properties

  • If an uncountable set X is a subset of set Y, then Y is uncountable.

Examples

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, such as the set of all infinite sequences of natural numbers and the set of all subsets of the set of natural numbers. The cardinality of R is often called the cardinality of the continuum and denoted by c, or 2^{\aleph_0}, or \beth_1 (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.

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