|
In mathematics, an uncountable or nondenumerable 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 is an injection from the natural numbers N to X, but no injection from X to N. For other meanings of mathematics or math, see mathematics (disambiguation). ...
In mathematics, a set can be thought of as any collection of distinct things considered as a whole. ...
In mathematics, a countable set is a set with the same cardinality (i. ...
In mathematics, a countable set is a set with the same cardinality (i. ...
In mathematics, a set is called finite if there is a bijection between the set and some set of the form {1, 2, ..., n} where is a natural number. ...
The infinity symbol ∞ in several typefaces The word infinity comes from the Latin infinitas or unboundedness. ...
It has been suggested that this article or section be merged with Logical biconditional. ...
In mathematics, an injective function (or one-to-one function or injection) is a function which maps distinct input values to distinct output values. ...
Equivalently, an uncountable set is one whose cardinality is strictly greater than  (aleph-null, the cardinality of the natural numbers). Cardinality refers to the size of a set; these are analyzed with the theory of cardinal numbers. Uncountable sets have many different cardinalities. In mathematics, the cardinality of a set is a measure of the number of elements of the set. There are two approaches to cardinality – one which compares sets directly using bijections, injections, and surjections, and another which uses cardinal numbers. ...
In the branch of mathematics known as set theory, the aleph numbers are a series of numbers used to represent the cardinality (or size) of infinite sets. ...
In mathematics, a natural number is either a positive integer (1, 2, 3, 4, ...) or a non-negative integer (0, 1, 2, 3, 4, ...). The former definition is generally used in number theory, while the latter is preferred in set theory and computer science. ...
In linguistics, cardinal numbers is the name given to number words that are used for quantity (one, two, three), as opposed to ordinal numbers, words that are used for order (first, second, third). ...
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  (beth-one). In mathematics, the set of real numbers, denoted R, is the set of all rational numbers and irrational numbers. ...
Cantors diagonal argument is a proof devised by Georg Cantor to demonstrate that the real numbers are not countably infinite. ...
In mathematics, a sequence is a list of objects (or events) arranged in a linear fashion, such that the order of the members is well defined and significant. ...
In mathematics, a natural number is either a positive integer (1, 2, 3, 4, ...) or a non-negative integer (0, 1, 2, 3, 4, ...). The former definition is generally used in number theory, while the latter is preferred in set theory and computer science. ...
A is a subset of B, and B is a superset of A. In mathematics, especially in set theory, a set A is a subset of a set B, if A is contained inside B. The relationship of one set being a subset of another is called inclusion. ...
In mathematics, the cardinality of the continuum is the cardinal number of the set of real numbers R (sometimes called the continuum). ...
In mathematics, the cardinality of the continuum is the cardinal number of the set of real numbers R (sometimes called the continuum). ...
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. The Cantor set, introduced by German mathematician Georg Cantor, is a remarkable construction involving only the real numbers between zero and one. ...
It has been suggested that Fractal animation be merged into this article or section. ...
In mathematics, the Hausdorff dimension is an extended non-negative real number (that is a number in the closed infinite interval [0, ∞]) associated to any metric space . ...
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-two) and is, indeed, larger than . In set theory and other branches of mathematics, ‭ב‬2 (pronounced beth two), or 2c (pronounced two to the power of c), is a certain cardinal number. ...
A much more abstract example of an uncountable set is the set of all countable ordinal numbers, denoted Ω. The cardinality of Ω is denoted  (aleph-one). It can be shown that is the smallest uncountable cardinal number. One might naturally wonder whether , the cardinality of the reals, is equal to or if it is strictly larger. The statement that  is called the continuum hypothesis. This hypothesis is now known to be independent of 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. Commonly, ordinal numbers, or ordinals for short, are numbers used to denote the position in an ordered sequence: first, second, third, fourth, etc. ...
In the branch of mathematics known as set theory, the aleph numbers are a series of numbers used to represent the cardinality (or size) of infinite sets. ...
In mathematics, the continuum hypothesis is a hypothesis about the possible sizes of infinite sets. ...
Set theory is the mathematical theory of sets, which represent collections of abstract objects. ...
The Zermelo-Fraenkel axioms of set theory (ZF) are the standard axioms of axiomatic set theory on which, together with the axiom of choice, all of ordinary mathematics is based in modern formulations. ...
The forgoing assumes the axiom of choice. Without the axiom of choice, there might exist cardinalities incomparable to aleph-null (such as the cardinality of a Dedekind-finite infinite set). These are not considered to be uncountable. In mathematics, the axiom of choice, or AC, is an axiom of set theory. ...
In mathematics, a set A is Dedekind-infinite if some proper subset B of A is equinumerous to A. Explicitly, this means that there is a bijective function from A onto some proper subset B of A. A set is Dedekind-finite if it is not Dedekind-infinite. ...
[edit]
See also
[edit] In linguistics, cardinal numbers is the name given to number words that are used for quantity (one, two, three), as opposed to ordinal numbers, words that are used for order (first, second, third). ...
In the branch of mathematics known as set theory, the aleph numbers are a series of numbers used to represent the cardinality (or size) of infinite sets. ...
In mathematics, the Hebrew letter (aleph) with various subscripts represents various infinite cardinal numbers (see aleph number). ...
In mathematics, a natural number is either a positive integer (1, 2, 3, 4, ...) or a non-negative integer (0, 1, 2, 3, 4, ...). The former definition is generally used in number theory, while the latter is preferred in set theory and computer science. ...
In mathematics, an injective function (or one-to-one function or injection) is a function which maps distinct input values to distinct output values. ...
External links - Proof that R is uncountable
|
Feeds |
Contact