# User Contributed Dictionary

### Etymology

From un- + countable.### Adjective

- 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.

- 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.

- 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

#### Derived terms

#### Translations

too many to be counted

mathematics: incapable of being enumerated by
natural numbers

- Czech: nespočetný
- Danish: overtællelig
- Dutch: ontelbaar
- Finnish: ylinumeroituva
- German: zahllos
- Italian: innumerabile
- Slovene: nešteven , neštevna , neštevno
- Swedish: ouppräknelig

linguistics: about a noun which cannot be
counted

- ttbc French: indénombrable (1) invariable (2)
- ttbc Persian: (bišomâr)
- ttbc Spanish: incontable (1)

# 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.

## External links

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