AskDefine | Define uncountable

User Contributed Dictionary

English

Etymology

From un- + countable.

Adjective

  1. 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.
  2. Incapable of being put into one-to-one correspondence with the natural numbers or any subset thereof.
    Cantor’s “diagonal proof” shows that the real numbers are uncountable.
  3. Describes a meaning of a noun that cannot be used freely with numbers or the indefinite article, and which therefore takes no plural form. Example: information.
    Many languages do not distinguish countable nouns from uncountable nouns.
    One meaning in law of the supposedly uncountable noun "information" is used in the plural and is countable.

Antonyms

Translations

too many to be counted
mathematics: incapable of being enumerated by natural numbers
linguistics: about a noun which cannot be counted

Extensive Definition

In mathematics, an uncountable set is an infinite set which is too big 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 natural numbers. The related term nondenumerable set is used by some authors as a synonym for "uncountable set" while other authors define a set to be nondenumerable if it is not an infinite countable set.

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 even the set of all infinite sequences consisting only of zeros and ones) 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^, 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-Frankel 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.

References

  • Halmos, Paul, Naive set theory. Princeton, NJ: D. Van Nostrand Company, 1960. Reprinted by Springer-Verlag, New York, 1974. ISBN 0-387-90092-6 (Springer-Verlag edition).
  • Jech, Thomas, 2003. Set Theory: The Third Millennium Edition, Revised and Expanded. Springer. ISBN 3-540-44085-2.
uncountable in Czech: Nespočetná množina
uncountable in German: Überabzählbarkeit
uncountable in French: Ensemble infini non dénombrable
uncountable in Italian: Insieme non numerabile
uncountable in Georgian: არათვლადი სიმრავლე
uncountable in Dutch: Overaftelbaarheid
uncountable in Polish: Zbiór nieprzeliczalny
uncountable in Portuguese: Conjunto não-enumerável
uncountable in Slovak: Nespočítateľná množina
uncountable in Finnish: Ylinumeroituva joukko
uncountable in Swedish: Överuppräknelig
uncountable in Tamil: எண்ணுறா முடிவிலிகள்
uncountable in Chinese: 不可數集
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