In mathematics the term countable is used to describe the size of a set, i.e. the number of elements it contains. The notion of an infinite set is not elementary; it requires a strong sense of abstraction and precision. A set is called countable if the number of elements is finite or if it has the same number of elements as the natural numbers. (Cantor defined a countable set as a set which can be put into one-to-one correspondence with a subset of the natural numbers). The term countable stems from the fac ...

Including:

Countable set - Definition Countable set - Gentle introduction
Countable set - A more formal introduction Countable set - Further theorems about uncountable sets

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Countable set, Countable set - Definition, Countable set - Gentle introduction, Countable set - A more formal introduction, Countable set - Further theorems about uncountable sets

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A set S is called countable if there exists an injective function If f is also bijective then S is called countably infinite or denumerable. As noted above, this terminology is not universal: some authors define denumerable to mean what we have called "countable"; some define countable to mean what we have called "countably infinite". The next result offers an alternative definition of a countable set ...

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Countable set, Countable set - Definition, Countable set - Gentle introduction, Countable set - A more formal introduction, Countable set - Further theorems about uncountable sets, Countable set - Mentionable and interesting numbers

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The elements of a finite set can be listed, say { a1, a2, ..., an}. However, insofar as a set is a logical description of the properties of its members, it need not be finite. To understand this, imagine that I ask you: how many words can you make out of Scrabble pieces if you are allowed to ask me for more pieces no matter how many you used up? The answer? As many as you like; you can go forever. But that doesn't mean they won't each of them be a word made out of scrabble b ...

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Countable set, Countable set - Definition, Countable set - Gentle introduction, Countable set - A more formal introduction, Countable set - Further theorems about uncountable sets

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The elements of a finite set can be listed, say { a1, a2, ..., an }. However, insofar as a set is a logical description of the properties of its members, it need not be finite. To understand this, imagine that I ask you: how many words can you make out of Scrabble pieces if you are allowed to ask me for more pieces no matter how many you used up? The answer? As many as you like; you can go forever. But that doesn't mean they won't each of them be a word made out of scrabble ...

See also:

Countable set, Countable set - Definition, Countable set - Gentle introduction, Countable set - A more formal introduction, Countable set - Further theorems about uncountable sets, Countable set - Mentionable and interesting numbers

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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. Cardinality - Comparing sets. We say that two sets A and B have the same cardinality if there exists a bijection, i.e. a injective and surjective function, from A to B. For example, the set E = {2, 4, 6, ...} of positi ...

Including:

Cardinality - Comparing sets Cardinality - Countable and uncountable sets Cardinality - Cardinal numbers Cardinality - Examples and other properties

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In mathematics, an axiom of countability is a property of certain mathematical objects (usually in a category) that requires the existence of a countable set with certain properties, while without it such sets might not exist. Important countability axioms for topological spaces: first-countable spaces: every point has a countable local base, second-countable spaces: the topology has a countable base, separable spaces: there exists a countable dense subspace, Lindelöf spaces: every open cover has a countable subcover, σ-compact spac ...

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Note that, up until this point, we have only defined the term "cardinality" in a strictly functional role: we have not actually defined the "cardinality" of a set as a specified object itself. We now outline such an approach. The relation of having the same cardinality is called equinumerosity, and this is an equivalence relation on the class of all sets. The equivalence class of a set A under this relation then consists of all those sets which have the same cardinality as A. There are then two main approaches to the def ...

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Cardinality, Cardinality - Comparing sets, Cardinality - Countable and uncountable sets, Cardinality - Cardinal numbers, Cardinality - Examples and other properties

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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 does not exist a surjective function from the natural numbers N to X. Not all uncountable sets have the same size; the sizes of infinite sets are analyzed with the theory of cardinal numbers. Formally, an uncountable set is defined as one whose cardinality is strictly greater than ...

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In epistemology, an axiom is a self-evident truth upon which other knowledge must rest, from which other knowledge is built up. Not all epistemologists agree that any axioms, understood in that sense, exist. In mathematics, an axiom is not necessarily a self-evident truth but rather, a formal logical expression used in a deduction to yield further results. Mathematics distinguishes two types of axioms: logical axioms and non-logical axioms. Axiom - Etymology. The word axiomIncluding:

Axiom - Etymology Axiom - Mathematics
Axiom - Logical axioms Axiom - Non-logical axioms Axiom - Role in mathematical logic Axiom - Further discussion

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In Zermelo-Fränkel set theory, Cantor's theorem states that the power set (set of all subsets) of any set A has a strictly greater cardinality than that of A. Cantor's theorem is obvious for finite sets, but surprisingly it holds true for infinite sets as well. In particular, the power set of a countably infinite set is un-countably infinite. To illustrate the validity of Cantor's theorem for infinite sets, just test an infinite set in the proof below. Cantor's theorem - The proof. Including:

Cantor's theorem - The proof Cantor's theorem - A detailed explanation of the proof when X is countably infinite Cantor's theorem - History

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In mathematics, the Hebrew letter (aleph) with various subscripts represents various infinite cardinal numbers (see aleph number). The second Hebrew letter (beth) is also used. To define the beth numbers, start by letting be the cardinality of countably infinite sets; for concreteness, take the set of natural numbers to be the typical case. Denote by P(A) the power set of A, i.e., the set of all subsets of A. Then define = the cardinality of the power set of A if is the cardina ...

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In mathematics and physics, continuous spectrum is, roughly speaking, a non-countable set of eigenvalues of an operator. An operator acting on a Hilbert space is said to have a continuous spectrum if its eigenvalues can be changed continuously. If the spectrum of an operator is not continuous, we say that it is has discrete spectrum. Some of the basic questions in spectral theory are to characterise the discrete spectrum and purely continuous spectrum, just as a measure, such as a probability measure, can typically ...

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The integers consist of the positive natural numbers (1, 2, 3, …), their negatives (−1, −2, −3, ...) and the number zero. Like the natural numbers, the integers form a countably infinite set. The set of all integers is usually denoted in mathematics by a boldface Z (or blackboard bold, ), which stands for Zahlen (German for "numbers"). The term rational integer is used, in algebraic number theory, to distinguish these 'ordinary' integers, in the rational numbers, from other concepts such as t ...

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Integer - Algebraic properties Integer - Order-theoretic properties Integer - Integers in computing Integer - Quotations

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In mathematics, a transcendental number is any real number that is not algebraic, that is, not the solution of a non-zero polynomial equation with integer (or, equivalently, rational) coefficients. It follows that all transcendental numbers are irrational. However, not all irrational numbers are transcendental; √2 is irrational but is a solution of the polynomial x2 - 2 = 0. The set of all transcendental numbers is uncountable. The proof is simple: Since the polynomials with integer coefficients are countabl ...

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Transcendental number - Proof that e is transcendental

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We say that two sets A and B have the same cardinality if there exists a bijection, i.e. a injective and surjective function, from A to B. For example, the set E = {2, 4, 6, ...} of positive even numbers has the same cardinality as the set N = {1, 2, 3, ...} of natural numbers, since the function f(n) = 2n is a bijection from N to E. We say that a set A has cardinality greater than or equal to the cardinality of B (and B has cardinality l ...

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Cardinality, Cardinality - Comparing sets, Cardinality - Countable and uncountable sets, Cardinality - Cardinal numbers, Cardinality - Examples and other properties

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Čech cohomology is a particular type of cohomology in mathematics. It is named for the mathematician Eduard Čech. Čech cohomology - Construction. Let X be a topological space with open cover U={Uα}α∈I, where I is a countable ordered set. We will simplify notation by writing intersections as Uα∩ Uβ = Uαβ, and so on for higher intersections. For every intersection Uα ...

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ÄŒech cohomology - Construction ÄŒech cohomology - Relation to other cohomology theories

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Cantor's diagonal argument is a proof devised by Georg Cantor to demonstrate that the real numbers are not countably infinite. (It is also called the diagonalization argument or the diagonal slash argument or the diagonal method.) The diagonal argument was not Cantor's first proof of the uncountability of the real numbers, but was published three years after his first proof. His original argument did not mention decimal expansions, nor any other numeral system. Since this technique was first used, si ...

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Cantor's diagonal argument - Real numbers Cantor's diagonal argument - Why this does not work on integers Cantor's diagonal argument - General sets

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List of general topology topics - Compactness and countability. Compact space Relatively compact subspace Heine-Borel theorem Tychonoff's theorem Finite intersection property Compactification Measure of non-compactness Paracompact space Locally compact space Compactly generated space Axiom of countability First-countable space Second-countable space Separable space Lindel ...

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List of general topology topics, List of general topology topics - Basic concepts, List of general topology topics - Limits, List of general topology topics - Topological properties, List of general topology topics - Compactness and countability, List of general topology topics - Connectedness, List of general topology topics - Separation axioms, List of general topology topics - Topological constructions, List of general topology topics - Examples, List of general topology topics - Uniform spaces, List of general topology topics - Metric spaces, List of general topology topics - Topology and order theory, List of general topology topics - Descriptive set theory, List of general topology topics - Dimension theory, List of general topology topics - Topological algebra, List of general topology topics - Combinatorial topology, List of general topology topics - Foundations of algebraic topology

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Such a property allows for the comparison of how many elements are contained in two or more sets without resorting to an intermediate set (viz. the natural numbers). Within the realm of uncountable sets, there exists a class of sets Y such that | Y | = c (cardinality of set of real numbers). Such sets are said to have "cardinality of the continuum." It can be proven that there exists no set X such that for any set Y, | Y | â ...

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Cardinality, Cardinality - Comparing sets, Cardinality - Countable and uncountable sets, Cardinality - Cardinal numbers, Cardinality - Examples and other properties

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