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Mathematical set that can be enumerated
mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is countable
Countable_set
Finite sets whose elements are all hereditarily finite sets
proves it to be a set also proves it to be countable. In 1937, Wilhelm Ackermann introduced an encoding of hereditarily finite sets as natural numbers
Hereditarily_finite_set
Concept in mathematics
countable choice or axiom of denumerable choice, denoted ACω, is an axiom of set theory that states that every countable collection of non-empty sets
Axiom_of_countable_choice
Property in descriptive set theory
mathematical field of 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
Perfect_set_property
Measurable set whose measure is zero
null set is a Lebesgue measurable set of real numbers that has measure zero. This can be characterized as a set that can be covered by a countable union
Null_set
Class of mathematical sets
{\displaystyle X} that contains both the empty set and the entire set X {\displaystyle X} , and is closed under countable union and complement. Then we can define
Borel_set
In set theory, a set is called hereditarily countable if it is a countable set of hereditarily countable sets. The inductive definition above is well-founded
Hereditarily_countable_set
Generalization of mass, length, area and volume
{\displaystyle X} is countable; and semifinite (without regard to whether X {\displaystyle X} is countable). (Thus, counting measure, on the power set P ( X ) {\displaystyle
Measure_(mathematics)
Countable union of closed sets
In mathematics, an Fσ set (pronounced F-sigma set) is a countable union of closed sets. The notation originated in French with F for fermé (French: closed)
Fσ_set
Set with an equinumerous proper subset
ZF) conditions: it has a countably infinite subset; there exists an injective map from a countably infinite set (say, N, the set of all natural numbers)
Dedekind-infinite_set
Broadest definition of sizes in integer-dimensional spaces
a way that is compatible with countable unions and other kinds of countable limits of sets. For example, every countable subset of the real line has Lebesgue
Lebesgue_measure
Generalization of "n-th" to infinite cases
discrete sets, so they are countable. Proof of first theorem: If P(α) = ∅ for some index α, then P′ is the countable union of countable sets. Therefore
Ordinal_number
Set that is not a finite set
In set theory, an infinite set is a set that is not a finite set. Infinite sets may be countable or uncountable. The set of natural numbers (whose existence
Infinite_set
Collection of mathematical objects
finite sets or countably infinite sets (sets of cardinality ℵ 0 {\displaystyle \aleph _{0}} ); some authors use "countable" to mean "countably infinite"
Set_(mathematics)
V=L Axiom of countability Every set is hereditarily countable Axiom of countable choice The product of a countable number of non-empty sets is non-empty
Glossary_of_set_theory
Standard system of axiomatic set theory
intuitiveness. The language's alphabet consists of: A countably infinite number of variables used for representing sets The logical connectives ¬ {\displaystyle \lnot
Zermelo–Fraenkel_set_theory
Index of articles associated with the same name
mathematics, an axiom of countability is a property of certain mathematical objects that asserts the existence of a countable set with certain properties
Axiom_of_countability
Mapping function
infinite number of sets. A σ-additive set function is a function that has the additivity property even for countably infinite many sets, that is, μ ( ⋃ n
Sigma-additive_set_function
Maximal proper filter
sets is a countable set. However, ZF with the ultrafilter lemma is too weak to prove that a countable union of countable sets is a countable set. The Hahn–Banach
Ultrafilter_on_a_set
Size of a set in mathematics
the set of even numbers { 2 , 4 , 6 , ⋯ } {\displaystyle \{2,4,6,\cdots \}} and the set of rational numbers are countable. Uncountable sets are those
Cardinality
Property of mathematical sets
cocountable subset of a set X {\displaystyle X} is a subset Y {\displaystyle Y} whose complement in X {\displaystyle X} is a countable set. In other words, Y
Cocountability
Mathematical concept
be extended to an arbitrary countable set A (e.g. the set of n-tuples of integers, the set of rational numbers, the set of formulas in some formal language
Arithmetical_set
Branch of mathematics that studies sets
Kronecker objected to Cantor's proofs that the algebraic numbers are countable, and that the transcendental numbers are uncountable, results now included
Set_theory
Set of elements common to all of some sets
A_{2}\cap A_{3}\cap \cdots } ". This last example, an intersection of countably many sets, is actually very common; for an example, see the article on σ-algebras
Intersection_(set_theory)
Set of the elements not in a given subset
In set theory, the complement of a set A, often denoted by A c {\displaystyle A^{c}} (or A′), is the set of elements not in A. When all elements in the
Complement_(set_theory)
Type of cardinal number in mathematics
_{1}} are countable (finite or denumerable). Assuming the axiom of choice, the union of a countable set of countable sets is itself countable. So ℵ 1 {\displaystyle
Regular_cardinal
Axiomatic set theories based on the principles of mathematical constructivism
The empty set is not inhabited but generally deemed countable too, and note that the successor set of any countable set is countable. The set ω {\displaystyle
Constructive_set_theory
Branch of topology
infinite set. Any set can be given the cocountable topology, in which a set is defined as open if it is either empty or its complement is countable. When
General_topology
Diagram that shows all possible logical relations between a collection of sets
between sets, popularized by John Venn (1834–1923) in the 1880s. The diagrams are used to teach elementary set theory, and to illustrate simple set relationships
Venn_diagram
Every countable set is a strong measure zero set, and so is every union of countably many strong measure zero sets. Every strong measure zero set has Lebesgue
Strong_measure_zero_set
Set with operations obeying given axioms
which may be considered an operator that takes zero arguments. Given a (countable) set of variables x, y, z, etc. the term algebra is the collection of all
Algebraic_structure
Concept in mathematical logic
set theories in which sets can be members of themselves. For example, a set that contains only itself is a hereditary set. Hereditarily countable set
Hereditary_set
"Small" subset of a topological space
topology, a meagre set (also called a meager set or a set of first category) is a subset of a topological space that is a countable union of subsets that
Meagre_set
Bounded sequence in finite-dimensional Euclidean space has a convergent subsequence
{\displaystyle I} denotes its index set) has a convergent subsequence if and only if there exists a countable set K ⊆ I {\displaystyle K\subseteq I} such
Bolzano–Weierstrass_theorem
Mathematical function for the probability a given outcome occurs in an experiment
x}p(\omega ).} The points where the cdf jumps always form a countable set; this may be any countable set and thus may even be dense in the real numbers. A discrete
Probability_distribution
Mathematical set containing no elements
the empty set or void set is the unique set having no elements; its size or cardinality (count of elements in a set) is zero. Some axiomatic set theories
Empty_set
Topology made of cocountable subsets
known as the countable complement topology, is a topology that can be defined on any infinite set X {\displaystyle X} . In this topology, a set is open if
Cocountable_topology
Ordered listing of items in collection
is sometimes used for countable sets. However it is also often used for computably enumerable sets, which are the countable sets for which an enumeration
Enumeration
Mathematician (1845–1918)
Cantor 1874 A countable set is a set which is either finite or denumerable; the denumerable sets are therefore the infinite countable sets. However, this
Georg_Cantor
Description of continuous random distribution
of discrete random variables (random variables that take values on a countable set), while the PDF is used in the context of continuous random variables
Probability_density_function
Algebraic structure of set algebra
a set X {\displaystyle X} is a nonempty collection Σ {\displaystyle \Sigma } of subsets of X {\displaystyle X} closed under complement, countable unions
Σ-algebra
Particular class of sets which can be described entirely in terms of simpler sets
H_{\alpha }} is the set of sets which are hereditarily of cardinality less than α {\displaystyle \alpha } (see hereditarily countable set#Generalizations)
Constructible_universe
Infinite set that is not countable
mathematics, an uncountable set, informally, is an infinite set that contains too many elements to be countable. The uncountability of a set is closely related
Uncountable_set
Finite collection of distinct objects
finite set is finite. All finite sets are countable, but not all countable sets are finite. (Some authors, however, use "countable" to mean "countably infinite"
Finite_set
In mathematics, a non-algebraic number
\mathbb {C} } are both uncountable, and therefore larger than any countable set. All transcendental real numbers (also known as real transcendental
Transcendental_number
Technique invented by Paul Cohen for proving consistency and independence results
within M {\displaystyle M} (e.g. the countability of M {\displaystyle M} ), and thus prove the existence of sets that are "too complex for M {\displaystyle
Forcing_(mathematics)
Axiom of set theory
numbers are countable: As pointed out above, to show that a countable union of countable sets is itself countable requires the Axiom of countable choice.
Axiom_of_choice
On convergent subsequences of functions that are locally of bounded total variation
}}y\to x\}} be the set of discontinuities of f n {\displaystyle f_{n}} ; each of these sets are countable by the above basic fact. The set A := ( ⋃ n ∈ N
Helly's_selection_theorem
Borel sets on spaces without a countable base for the topology. In practice, the use of Baire measures on Baire sets can often be replaced by the use
Baire_set
Set of elements in any of some sets
In set theory, the union (denoted by ∪) of a collection of sets is the set of all elements in the collection. It is one of the fundamental operations
Union_(set_theory)
Set which cannot be assigned a meaningful "volume"
formulate probability theory on sets which are constrained to be measurable. The measurable sets on the line are iterated countable unions and intersections
Non-measurable_set
Process of mapping a continuous set to a countable set
of mapping input values from a large set (often a continuous set) to output values in a (countable) smaller set, often with a finite number of elements
Quantization (signal processing)
Quantization_(signal_processing)
Set of all limit points of a set
states that any Polish space can be written as the union of a countable set and a perfect set. Because any Gδ subset of a Polish space is again a Polish
Derived_set_(mathematics)
related to set theory. Algebra of sets Axiom of choice Axiom of countable choice Axiom of dependent choice Zorn's lemma Axiom of power set Boolean-valued
List_of_set_theory_topics
Mathematical lemma
is a countable set of dense subsets of P then there exists a D-generic filter F in P such that p ∈ F. Let p ∈ P be given. Since D is countable, D = { Di |
Rasiowa–Sikorski_lemma
First article on transfinite set theory
theorems is his "revolutionary discovery" that the set of all real numbers is uncountably, rather than countably, infinite. This theorem is proved using Cantor's
Cantor's first set theory article
Cantor's_first_set_theory_article
Minimal standard model of ZFC
Löwenheim–Skolem theorem implies that the minimal model (if it exists as a set) is a countable set. More precisely, every element s of the minimal model can be named;
Minimal_model_(set_theory)
Use of braces for specifying sets
{Z} ,n=2k\}} — The set of all even integers, expressed in set-builder notation. In mathematics and more specifically in set theory, set-builder notation
Set-builder_notation
Topological space whose topology has a countable base
topology, a second-countable space, also called a completely separable space, is a topological space whose topology has a countable base. More explicitly
Second-countable_space
Geometric theorem
G-equidecomposable sets may be found whose "sizes" vary. Moreover, since a countable set can be made into two copies of itself, one might expect that using countably many
Banach–Tarski_paradox
Informal set theories
Naive set theory is any of several set theories used in the discussion of the foundations of mathematics. Unlike axiomatic set theories, which are defined
Naive_set_theory
Study of discrete mathematical structures
characterized as the branch of mathematics dealing with countable sets (finite sets or sets with the same cardinality as the natural numbers). However
Discrete_mathematics
Any collection of sets, or subsets of a set
sets δ-ring – Ring closed under countable intersections Field of sets – Algebraic concept in measure theory, also referred to as an algebra of sets Generalized
Family_of_sets
complexity, and showing (under certain assumptions) that there are largest countable sets of certain complexities. Given a pointset A contained in some product
Scale (descriptive set theory)
Scale_(descriptive_set_theory)
Paradox in set theory
existence of countable models (Skolem's paradox), but it enjoys some important advantages." In ZFC, given a set A, it is possible to define a set B that consists
Russell's_paradox
Infinite cardinal number
(this follows from the fact that the union of a countable number of countable sets is itself countable). This fact is analogous to the situation in ℵ 0
Aleph_number
Smallest ordinal number that, considered as a set, is uncountable
the order type of an uncountable well-ordered set. It is the supremum (least upper bound) of all countable ordinals. In the von Neumann representation,
First_uncountable_ordinal
Monotone maps have countable discontinuities
(monotone) function are necessarily jump discontinuities and there are at most countably many of them. Usually, this theorem appears in literature without a name
Discontinuities of monotone functions
Discontinuities_of_monotone_functions
{\displaystyle X} is called countably generated if the topology of X {\displaystyle X} is determined by the countable sets in a similar way as the topology
Countably_generated_space
Concept in topology
space; that is, a space homeomorphic to a complete metric space that has a countable dense subset. Polish spaces are so named because they were first extensively
Polish_space
Any one of the distinct objects that make up a set in set theory
mathematics, an element (or member) of a set is any one of the distinct objects that belong to that set. For example, given a set called A containing the first four
Element_of_a_set
Inequality applying to probability spaces
inequality, also known as the union bound, says that for any finite or countable set of events, the probability that at least one of the events happens is
Boole's_inequality
Sets whose elements have degrees of membership
In mathematics, fuzzy sets are sets whose elements have degrees of membership. Fuzzy sets were introduced independently by Lotfi A. Zadeh in 1965 as an
Fuzzy_set
convert any standard (set) model of ZFC into a standard transitive model M that is itself countable. Every set in M must be countable in V, but at the same
Standard_model_(set_theory)
Problem of finding the best feasible solution
object such as an integer, permutation or graph must be found from a countable set. A problem with continuous variables is known as a continuous optimization
Optimization_problem
Point of a subset S around which there are no other points of S
injectively onto a set of points with rational coordinates, of which there are only countably many. However, not every countable set is discrete, of which
Isolated_point
Area of mathematical logic
quasiminimally excellent classes are those in which every definable set is either countable or co-countable. They are key to the model theory of the complex exponential
Model_theory
Declarative logic programming language
If constant and variable are two countable sets of constants and variables respectively and relation is a countable set of predicate symbols, then the following
Datalog
Uniqueness of countable dense linear orders
mathematics, Cantor's isomorphism theorem states that every two nonempty countable dense unbounded linear orders are order-isomorphic. The theorem is named
Cantor's_isomorphism_theorem
Number used for counting
product of primes Countable set – Mathematical set that can be enumerated Sequence – Function of the natural numbers in another set Ordinal number – Generalization
Natural_number
Size of a possibly infinite set
infinite sets (for example the set of natural numbers N = {0, 1, 2, 3, ...}). Thus, he called all sets having a bijection with N denumerable (countably infinite)
Cardinal_number
Philosphical view that existence proofs must be constructive
then opens the question as to what sort of function from a countable set to a countable set, such as f and g above, can actually be constructed. Different
Constructivism (philosophy of mathematics)
Constructivism_(philosophy_of_mathematics)
Anticommutating number
finite number of generators, typically n = 1, 2, 3 or 4, and those with a countably-infinite number of generators. These two situations are not as unrelated
Grassmann_number
Sets can be classified according to the properties they have. Empty set Finite set, Infinite set Countable set, Uncountable set Power set Closed set Open
List_of_types_of_sets
Mathematical logic concept
is the apparent contradiction that a countable model of first-order set theory could contain an uncountable set. The paradox arises from part of the Löwenheim–Skolem
Skolem's_paradox
Set of points on a line segment with certain topological properties
set (equipped with its subspace topology). The Cantor set is naturally homeomorphic to the countable product 2 _ N {\displaystyle {\underline {2}}^{\mathbb
Cantor_set
Expected value of a random variable given that certain conditions are known to occur
{\mathcal {H}})} may have a different null set. Because countable unions of null sets are null sets, for a countable set of X i {\displaystyle X_{i}} , one can
Conditional_expectation
Set whose elements all belong to another set
In mathematics, a set A is a subset of a set B if and only if all elements of A are also elements of B; B is then a superset of A. It is possible for A
Subset
Kind of proof calculus
{\displaystyle {\mathcal {T}}} . We shall fix a countable set V {\displaystyle V} of variables, another countable set F {\displaystyle F} of function symbols
Natural_deduction
to a countable set of values. In his doctoral dissertation, Wetzel asked whether this assumption implies that F is necessarily itself countable. Paul
Wetzel's_problem
Lebesgue measure) if it is concentrated on an at most countable set. The support need not be a discrete set. Geometrically, a discrete measure (on the real
Discrete_measure
Branch of mathematical logic
has an algebraic closure" is not provable in ZF set theory, but the restricted form "every countable field has an algebraic closure" is provable in RCA0
Reverse_mathematics
Topics referred to by the same term
freedom (physics and chemistry), a concept describing dependence on a countable set of parameters Degree of frost, a unit of temperature measurement Degrees
Degree
Subset that is closed and has no isolated points
can be uniquely written as the disjoint union of a perfect set and a countable set. This is also true more generally for all closed subsets of Polish spaces
Perfect_set
Identities and relationships involving sets
algebra of sets, completed to include countably infinite operations. Axiomatic set theory Image (mathematics) § Properties Field of sets List of set identities
Algebra_of_sets
Collection of random variables
time, if the index set of a stochastic process has a finite or countable number of elements, such as a finite set of numbers, the set of integers, or the
Stochastic_process
the same size (cardinality) as N, and is said to be countable. Examples of countably infinite sets are the natural numbers, the even numbers, the prime
Paradoxes_of_set_theory
Countable intersection of open sets
set is a subset of a topological space that is a countable intersection of open sets. The notation originated from the German nouns Gebiet 'open set'
Gδ_set
Function that is continuous everywhere but differentiable nowhere
visualise a continuous function whose set of nondifferentiable points is something other than a countable set of points. Analogous results for better
Weierstrass_function
Model of computation over real numbers
latter are by definition restricted to a finite set of symbols. A Turing machine can represent a countable set (such as the rational numbers) by strings of
Blum–Shub–Smale_machine
COUNTABLE SET
COUNTABLE SET
Boy/Male
Shakespearean
Much Ado About Nothing' A Constable.
Male
Greek
(Σήθι) Greek form of Egyptian Seti, SETHI means "of Seth."Â
Male
English
Anglicized form of Hebrew Sheth, SETH means "buttocks." In the bible, this is the name of the third son of Adam and Eve. Compare with other forms of Seth.
Surname or Lastname
English
English : occupational name for a stone- or bricklayer, from Middle English setter ‘one who lays stones or bricks in building’ (agent derivative of setten ‘to set’).English : occupational name from Old French saietier ‘silk weaver’ (an agent derivative of sayete, a kind of silk).English : from an agent derivative of Middle English setten ‘to place (decoration, on a garment or metal surface)’, probably an occupational name for an embroiderer.German : unexplained.Norwegian : unexplained.
Boy/Male
Norse
Pointable.
Boy/Male
Assamese, Celebrity, Gujarati, Hindu, Indian, Jain, Kannada, Malayalam, Marathi, Oriya, Punjabi, Sanskrit, Sikh, Tamil, Telugu, Traditional
Sky; Lord of Day; Uncountable; Space
Male
Hindi/Indian
(सेठ) Hindi name derived from the Sanskrit word setu, SETH means "bridge." Compare with other forms of Seth.
Surname or Lastname
English and French (Châtelain)
English and French (Châtelain) : status name for the governor or constable of a castle, or the warder of a prison, from Norman Old French chastelain (Latin castellanus, a derivative of castellum ‘castle’).A priest named Châtelain from Paris is documented in Quebec city in 1636, and a family is documented in Trois Rivières, Quebec, in 1722.
Male
Italian
Italian form of Roman Latin Septimus, SETTIMIO means "seventh."
Boy/Male
Gujarati, Hindu, Indian, Kannada, Malayalam, Marathi, Oriya, Punjabi, Sikh, Sindhi, Tamil, Telugu
Brave; Winner; Smart; Strong; Uncountable; Infinite God
Boy/Male
Shakespearean
Measure for Measure' A simple constable.
Surname or Lastname
English
English : occupational name for the law-enforcement officer of a parish, from Middle English, Old French conestable, cunestable, from Late Latin comes stabuli ‘officer of the stable’. The title was also borne by various other officials during the Middle Ages, including the chief officer of the household (and army) of a medieval ruler, and this may in some cases be the source of the surname.Americanized spelling of Dutch Constapel, an occupational name for the chief gunner aboard a ship or in the garrison of a fort.
Boy/Male
Hindu, Indian
Uncountable
Female
Japanese
(節å) Japanese name SETSUKO means "temperate child."
Surname or Lastname
English
English : patronymic from Setter.
Boy/Male
Shakespearean
Love's Labours Lost' A constable.
Boy/Male
Hindu, Indian
Un Countable; Multiple; Countless
Male
Greek
(Σήθος) Greek form of Egyptian Sutekh, possibly SETHOS means "one who dazzles." In mythology, this is the name of an ancient evil god of Chaos, storms, and the desert, who slew Osiris.Â
Surname or Lastname
English
English : habitational name from a place in North Yorkshire, so named from Old English setl ‘seat’, ‘dwelling’.
Boy/Male
Hindu, Indian
Uncountable
COUNTABLE SET
COUNTABLE SET
Girl/Female
Arabic, Muslim
Increase; Excess
Boy/Male
Indian
The victorious
Male
Welsh
Welsh surname transferred to forename use, derived from ap Rhys, PRYCE means "son of Rhys."
Girl/Female
Hindu
Goddess Saraswati
Girl/Female
French, German
Little and Womanly
Girl/Female
Gujarati, Hindu, Indian, Kannada, Malayalam, Sanskrit
Entire; All; Earth; World; Universe
Girl/Female
Hindu
Male
Arthurian
, (Sir), christened Saracen knight; loved Isolde.
Boy/Male
Sikh
Boy/Male
Latin
Son of Azeus.
COUNTABLE SET
COUNTABLE SET
COUNTABLE SET
COUNTABLE SET
COUNTABLE SET
n.
A peace officer; an under constable.
a.
Liable to be called on to render an account; answerable; as, every man is accountable to God for his conduct.
n.
An under constable.
n.
The office or functions of a constable.
v. t.
To submit; to make accountable.
n.
The quality or state of being accountable; accountability.
a.
Capable of being numbered.
n.
The quality of being cogitable; conceivableness.
a.
Not accountable or responsible; free from control.
a.
Such as can be mounted.
adv.
In an accountable manner.
n.
The wife of a constable.
a.
Not cogitable; inconceivable.
n.
One who renders account; one accountable.
v. t.
Accountable; responsible; sensitive.
a.
See Accountable.
a.
Accountable.
a.
Capable of being thought or conceived; cogitable.
n.
The state or quality of being numerable or countable.
n.
A petty constable.