Uncountable set

Uncountable set

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 every 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 proven 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.

Another example of an uncountable set is the set of all functions from R to R. This set is even "more uncountable" than R in the sense that the cardinality of this set is \beth_2 (beth-two), which is larger than \beth_1.

A more abstract example of an uncountable set is the set of all countable ordinal numbers, denoted by Ω (omega) or ω1. The cardinality of Ω is denoted \aleph_1 (aleph-one). It can be shown, using the axiom of choice, that \aleph_1 is the smallest uncountable cardinal number. Thus either \beth_1, the cardinality of the reals, is equal to \aleph_1 or it is strictly larger. Georg Cantor was the first to propose the question of whether \beth_1 is equal to \aleph_1. In 1900, David Hilbert posed this question as the first of his 23 problems. The statement that \aleph_1 = \beth_1 is now called the continuum hypothesis and is known to be independent of the Zermelo–Fraenkel axioms for set theory (including the axiom of choice).

Without the axiom of choice

Without the axiom of choice, there might exist cardinalities incomparable to \aleph_0 (namely, the cardinalities of Dedekind-finite infinite sets). Sets of these cardinalities satisfy the first three characterizations above but not the fourth characterization. Because these sets are not larger than the natural numbers in the sense of cardinality, some may not want to call them uncountable.

If the axiom of choice holds, the following conditions on a cardinal \kappa\! are equivalent:

  • \kappa \nleq \aleph_0;
  • \kappa > \aleph_0; and
  • \kappa \geq \aleph_1, where \aleph_1 = |\omega_1 | and \omega_1\, is least initial ordinal greater than \omega.\!

However, these may all be different if the axiom of choice fails. So it is not obvious which one is the appropriate generalization of "uncountability" when the axiom fails. It may be best to avoid using the word in this case and specify which of these one means.

See also

References

External links


Wikimedia Foundation. 2010.

Игры ⚽ Нужна курсовая?

Look at other dictionaries:

  • uncountable set — noun A set, containing infinite number of elements, whose elements can not be mapped one to one to the natural numbers. A set with a cardinality greater than that of the set of natural numbers …   Wiktionary

  • Set (mathematics) — This article gives an introduction to what mathematicians call intuitive or naive set theory; for a more detailed account see Naive set theory. For a rigorous modern axiomatic treatment of sets, see Set theory. The intersection of two sets is… …   Wikipedia

  • Set theory — This article is about the branch of mathematics. For musical set theory, see Set theory (music). A Venn diagram illustrating the intersection of two sets. Set theory is the branch of mathematics that studies sets, which are collections of objects …   Wikipedia

  • Set theory of the real line — is an area of mathematics concerned with the application of set theory to aspects of the real numbers. For example, one knows that all countable sets of reals are null, i.e. have Lebesgue measure 0; one might therefore ask the least possible size …   Wikipedia

  • uncountable — 1. noun An uncountable noun. 2. adjective a) So many as to be incapable of being counted. The reasons for our failure were as uncountable as the grains of sand on a beach. b) Incapable of being put into one to one …   Wiktionary

  • SET — See: Securities Exchange of Thailand See: Stock Exchange of Thailand * * * SET SET noun [uncountable] COMPUTING secure electronic transfer a way of buying and paying for goods on the Internet that allows the safe exchange of personal and… …   Financial and business terms

  • set-aside — UK / US noun [uncountable] British a) a system in which the European Union pays farmers not to grow crops in particular areas, in order to control prices or the amount of crops grown b) land that is not used for crops according to this system …   English dictionary

  • Countable set — Countable redirects here. For the linguistic concept, see Count noun. Not to be confused with (recursively) enumerable sets. In mathematics, a countable set is a set with the same cardinality (number of elements) as some subset of the set of… …   Wikipedia

  • Luzin set — In real analysis and descriptive set theory, a Luzin set (or Lusin set), named for N. N. Luzin, is an uncountable subset A of the reals such that every uncountable subset of A is nonmeager; that is, of second Baire category. Equivalently, A is an …   Wikipedia

  • Perfect set property — In descriptive set theory, a subset of a Polish space has the perfect set property if it is either countable or has a nonempty perfect subset.As nonempty perfect sets in a Polish space always have the cardinality of the continuum, a set with the… …   Wikipedia

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”