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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
Proof in set theory
elements than there are positive integers. Such sets are now called uncountable sets, and the size of infinite sets is treated by the theory of cardinal numbers
Cantor's_diagonal_argument
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
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
Mathematical set that can be enumerated
Cantor, who proved the existence of uncountable sets, that is, sets that are not countable; for example the set of the real numbers R {\displaystyle
Countable_set
First article on transfinite set theory
that the set of all real numbers is uncountably, rather than countably, infinite. This theorem is proved using Cantor's first uncountability proof, which
Cantor's first set theory article
Cantor's_first_set_theory_article
Property in descriptive set theory
a set of reals with the perfect set property cannot be a counterexample to the continuum hypothesis, stated in the form that every uncountable set of
Perfect_set_property
Set of elements common to all of some sets
geometric sets of points, such as individual points, lines (infinite uncountable sets of points), planes, etc. Intersection is written using the symbol "
Intersection_(set_theory)
Fractal named after mathematician Benoit Mandelbrot
study of the Mandelbrot set remains a key topic in the field of complex dynamics. The Mandelbrot set is the uncountable set of values of c in the complex
Mandelbrot_set
Smallest ordinal number that, considered as a set, is uncountable
is the smallest ordinal number that is the order type of an uncountable well-ordered set. It is the supremum (least upper bound) of all countable ordinals
First_uncountable_ordinal
Set of points on a line segment with certain topological properties
Cantor set a universal probability space in some ways. In Lebesgue measure theory, the Cantor set is an example of a set which is uncountable and has
Cantor_set
Collection of mathematical objects
"countably infinite". Sets with cardinality strictly greater than ℵ 0 {\displaystyle \aleph _{0}} are called uncountable sets. Cantor's diagonal argument
Set_(mathematics)
Set of real numbers that is not Lebesgue measurable
existence theorem that there are such sets. Each Vitali set is uncountable, and there are uncountably many Vitali sets. The proof of their existence depends
Vitali_set
Fractal sets in complex dynamics of mathematics
{\displaystyle \operatorname {J} (f)} is a nowhere dense set (it is without interior points) and an uncountable set (of the same cardinality as the real numbers)
Julia_set
Measurable set whose measure is zero
as subsets of the real numbers. The Cantor set is an example of an uncountable null set. It is uncountable because it contains all real numbers between
Null_set
Mathematical logic concept
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
Branch of mathematics that studies sets
sizes of two sets by setting them in one-to-one correspondence. His "revolutionary discovery" was that the set of all real numbers is uncountable, that is
Set_theory
Broadest definition of sizes in integer-dimensional spaces
being a countable union of points, which have no length, while many uncountable sets also have measure zero. The measure is not defined on every subset
Lebesgue_measure
Property in general topology
terms of closed sets; this is its most prominent application. Other applications include proving that certain perfect sets are uncountable, and the construction
Finite_intersection_property
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
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 mathematical space
the lower limit topology, no uncountable set is compact. In the cocountable topology on an uncountable set, no infinite set is compact. Like the previous
Compact_space
strong measure zero set has Lebesgue measure 0. The Cantor set is an example of an uncountable set of Lebesgue measure 0 which is not of strong measure zero
Strong_measure_zero_set
Standard system of axiomatic set theory
In set theory, Zermelo–Fraenkel set theory, named after mathematicians Ernst Zermelo and Abraham Fraenkel, is an axiomatic system that was proposed in
Zermelo–Fraenkel_set_theory
Axiomatic set theories based on the principles of mathematical constructivism
relations involving uncountable sets are also elusive in Z F C {\displaystyle {\mathsf {ZFC}}} , where the characterization of uncountability simplifies to
Constructive_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
Type of probability distribution
infinite set of components is covered formally by allowing n = ∞ {\displaystyle n=\infty \!} . Where the set of component distributions is uncountable, the
Mixture_distribution
Concept in set theory
be the set of Gödel numbers of the true sentences about the constructible universe, with c i {\displaystyle c_{i}} interpreted as the uncountable cardinal
Zero_sharp
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
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)
Identities and relationships involving sets
mathematics, particularly in the study of set theory, the algebra of sets defines the properties and laws of sets, the set-theoretic operations of union, intersection
Algebra_of_sets
Type of topological space
cofinite topology defined on an infinite set, as is the cocountable topology defined on an uncountable set. Pseudometric spaces typically are not Hausdorff
Hausdorff_space
Mathematical term
\end{cases}}} The set of all such indicator functions, { 1 r } r ∈ R {\displaystyle \{\mathbf {1} _{r}\}_{r\in \mathbb {R} }} , is an uncountable set indexed by
Index_set
Class of mathematical sets
to the first uncountable ordinal. To prove this claim, any open set in a metric space is the union of an increasing sequence of closed sets. In particular
Borel_set
Book by Lynn Steen
second-countable is counterexample #3, the discrete topology on an uncountable set. This particular counterexample shows that second-countability does
Counterexamples_in_Topology
Paradox in set theory
a set-theoretic paradox published by the British philosopher and mathematician, Bertrand Russell, in 1901. Russell's paradox shows that every set theory
Russell's_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
Point of a subset S around which there are no other points of S
explicit set consisting entirely of isolated points but has the counter-intuitive property that its closure is an uncountable set. Another set F with the
Isolated_point
Mathematician (1845–1918)
and infinite sets, subdividing the latter into denumerable (or countably infinite) sets and nondenumerable sets (uncountably infinite sets). Cantor developed
Georg_Cantor
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
Generalization of "n-th" to infinite cases
Thus, P(β) is a perfect set, so it is uncountable. Since P(β) ⊆ P′, the set P′ is uncountable. In both cases, P′ is uncountable, which contradicts P′ being
Ordinal_number
Infinite cardinal number
itself an ordinal number larger than all countable ones, so it is an uncountable set. Therefore, ℵ 1 {\displaystyle \aleph _{1}} is the smallest cardinality
Aleph_number
Generalization of mass, length, area and volume
(Thus, counting measure, on the power set P ( X ) {\displaystyle {\cal {P}}(X)} of an arbitrary uncountable set X , {\displaystyle X,} gives an example
Measure_(mathematics)
Topology made of cocountable subsets
compact nor countably metacompact, hence not compact. Uncountable set: On any uncountable set, such as the real numbers R {\displaystyle \mathbb {R}
Cocountable_topology
Frameworks for modeling variables that evolve over time
connected interval of the reals). That is, the function's domain is an uncountable set. The function itself need not to be continuous. To contrast, a discrete-time
Discrete time and continuous time
Discrete_time_and_continuous_time
Term in set theory
countable subset of the set of real numbers (which is uncountable). The Cantor set is uncountably infinite, but has Lebesgue measure zero. So almost all
Almost
Cantor's theorem is that the set of all real numbers R cannot be enumerated by natural numbers, that is, R is uncountable: |R| > |N|. Instead of relying
Paradoxes_of_set_theory
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
Three-dimensional fractal
Lebesgue measure 0. Because it contains continuous paths, it is an uncountable set. Experiments also showed that cubes with a Menger sponge-like structure
Menger_sponge
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
Area of mathematical logic
Skolem's paradox, since there are sentences in set theory which postulate the existence of uncountable sets and yet these sentences are true in our countable
Model_theory
Non-orientable surface with one edge
and cylinders, for which it is possible to simultaneously embed an uncountable set of disjoint copies into three-dimensional space, only a countable number
Möbius_strip
Set theory concept
In set theory and related branches of mathematics, the von Neumann universe, or von Neumann hierarchy of sets, denoted by V, is the class of hereditary
Von_Neumann_universe
Set of all things that may be the input of a mathematical function
In mathematics, the domain of a function is the set of inputs accepted by the function. It is sometimes denoted by dom ( f ) {\displaystyle \operatorname
Domain_of_a_function
Lκ is a set of indiscernibles for Lκ for every uncountable cardinal κ simply infinite set A term sometimes used for infinite sets, i.e., sets equinumerous
Glossary_of_set_theory
Topological space with a dense countable subset
example of an uncountable separable space is the real line, in which the rational numbers form a countable dense subset. Similarly the set of all length-
Separable_space
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
Theorem in combinatorial set theory extending Ramsey's theorem to uncountable sets
combinatorial set theory, a branch of mathematics, the Erdős–Rado theorem is a basic result extending Ramsey's theorem to uncountable sets. It is named
Erdős–Rado_theorem
Mathematical set formed from two given sets
In mathematics, specifically set theory, the Cartesian product of two sets A and B, denoted A × B, is the set of all ordered pairs (a, b) where a is an
Cartesian_product
Mathematical set containing all objects
In set theory, a universal set is a set that contains all of the objects in the theory, including itself. In set theory as usually formulated, it can
Universal_set
Mathematical set of all subsets of a set
countably infinite set is uncountably infinite. The power set of the set of natural numbers can be put in a one-to-one correspondence with the set of real numbers
Power_set
core model and satisfies the covering property, that is for every uncountable set x of ordinals, there is y such that y ⊃ x, y has the same cardinality
Covering_lemma
Any collection of sets, or subsets of a set
"family of sets" because if one instead uses "set of sets" then the subsequent use of "set" can be confusing as to whether it is the containing set or one
Family_of_sets
Collection of sets in mathematics that can be defined based on a property of its members
In set theory and its applications throughout mathematics, a class is a collection of mathematical objects (often sets) that can be unambiguously defined
Class_(set_theory)
set theory, a Luzin set (or Lusin set), is defined as an uncountable subset A of the reals R {\displaystyle \mathbb {R} } such that every uncountable
Luzin_space
Number that is not a ratio of integers
the reals form an uncountable set, of which the rationals are a countable subset, the complementary set of irrationals is uncountable. Under the usual
Irrational_number
Mathematical concept
If Ω is uncountable, still, it may happen that P(ω) ≠ 0 for some ω; such ω are called atoms. They are an at most countable (maybe empty) set, whose probability
Probability_space
refer to the assertion that every uncountable set of real numbers can be placed in bijective correspondence with the set of all reals. This second assertion
Weak_continuum_hypothesis
Sierpiński set is an uncountable subset of a real vector space whose intersection with every measure-zero set is countable. The existence of Sierpiński sets is
Sierpiński_set
Finite sets whose elements are all hereditarily finite sets
mathematics and set theory, hereditarily finite sets are defined as finite sets whose elements are all hereditarily finite sets. In other words, the set itself
Hereditarily_finite_set
Finite collection of distinct objects
(or the cardinal number) of the set. A set that is not a finite set is called an infinite set. For example, the set { 1 , 2 , 3 , … } {\displaystyle
Finite_set
of an uncountable set of functions that maps each argument x to a countable set of values is equivalent to the nonexistence of an uncountable set of real
Wetzel's_problem
Ordered listing of items in collection
Cantor's first uncountability proof. There exists an enumeration for a set (in this sense) if and only if the set is countable. If a set is enumerable
Enumeration
Measure of the "size" of linear operators
But { P t : 0 < t ≤ 1 } {\displaystyle \{P_{t}:0<t\leq 1\}} is an uncountable set. This implies the space of bounded operators on L 2 ( [ 0 , 1 ] ) {\displaystyle
Operator_norm
set and 0 otherwise. This function is 0 for an uncountable set of x values, and 1 for an uncountable set. It is discontinuous wherever it equals 1 and
Baire_function
Set with exactly one element
a singleton (also known as a unit set or one-point set) is a set with exactly one element. For example, the set { 0 } {\displaystyle \{0\}} is a singleton
Singleton_(mathematics)
Set with an equinumerous proper subset
The image of f is the countable set {f(n) | n ∈ N}, whose members are themselves infinite (and possibly uncountable) sets. By using the axiom of countable
Dedekind-infinite_set
Mathematical description of quantum state
are linearly independent, there are too many of them (they form an uncountable set) for a basis for physical Hilbert space. They can still be used to
Wave_function
Type of theory in mathematical logic
some uncountable cardinality, then it is categorical in all uncountable cardinalities. Saharon Shelah (1974) extended Morley's theorem to uncountable languages:
Categorical_theory
Partial order with well-ordered predecessors
hand, it is a theorem of ZFC that there are uncountable trees with no uncountable branches and no uncountable levels; such trees are known as Aronszajn
Tree_(set_theory)
Cardinality of the set of real numbers
cardinality to compare the sizes of infinite sets. He famously showed that the set of real numbers is uncountably infinite. That is, c {\displaystyle {\mathfrak
Cardinality_of_the_continuum
Mathematician and physicist
A set S is called a scrambled set if every pair of distinct points in S is scrambled. Scrambling is a kind of mixing. (2) There is an uncountably infinite
James_A._Yorke
All-encompassing set or class
In mathematics, and particularly in set theory, category theory, type theory, and the foundations of mathematics, a universe is a collection that contains
Universe_(mathematics)
Finding the number of elements of a finite set
a bijection with the natural numbers, and these sets are called "uncountable". Sets for which there exists a bijection between them are said to have the
Counting
Inputs for which a function's value is non-zero
{N} }} of functions from the natural numbers to the integers is the uncountable set of integer sequences. The subfamily { f ∈ Z N : f has finite support
Support_(mathematics)
Topological space where each point has a countable neighbourhood basis
topology on an uncountable set (such as the real line). More generally, the Zariski topology on an algebraic variety over an uncountable field is not first-countable
First-countable_space
System of mathematical set theory
prove the existence of ordinals α ≥ ω + ω, which include uncountably many hereditarily countable sets. This follows from Skolem's result that Vω+ω satisfies
Von Neumann–Bernays–Gödel set theory
Von_Neumann–Bernays–Gödel_set_theory
Elements in exactly one of two sets
symmetric difference of two sets, also known as the disjunctive union and set sum, is the set of elements which are in either of the sets, but not in their intersection
Symmetric_difference
Topological space that is homeomorphic to a metric space
exist metric spaces that are not second countable, for example, an uncountable set endowed with the discrete metric. The Nagata–Smirnov metrization theorem
Metrizable_space
System of mathematical set theory
of mathematics, Morse–Kelley set theory (MK), Kelley–Morse set theory (KM), Morse–Tarski set theory (MT), Quine–Morse set theory (QM) or the system of
Morse–Kelley_set_theory
Mathematical set regarded as insignificant
useful notion. Or let X be an uncountable set, and let a subset of X be negligible if it is countable. Then the negligible sets form a sigma-ideal. Let X
Negligible_set
Mathematical logic concept
In computability theory, a set S of natural numbers is called computably enumerable (c.e.), recursively enumerable (r.e.), semidecidable, partially decidable
Computably_enumerable_set
Set theory concept
ordinals). The set of all countable limit ordinals is a club set with respect to the first uncountable ordinal; but it is not a club set with respect to
Club_set
Infinite Cardinal number
with uncountably many discontinuities the power set of the set of all functions from the set of natural numbers to itself, or the number of sets of sequences
Beth_number
Means of constructing a group from two subgroups
an infinite (perhaps uncountable) set of subgroups, more care is needed. If g is an element of the cartesian product Π{Hi} of a set of groups, let gi be
Direct_sum_of_groups
Numeric quantity representing the center of a collection of numbers
circumstances, mathematicians may calculate a mean of an infinite (or even an uncountable) set of values. This can happen when calculating the mean value y avg {\displaystyle
Mean
Bernstein set is a subset of the real line that meets every uncountable closed subset of the real line but that contains none of them. A Bernstein set partitions
Bernstein_set
Axiom of set theory
countable while S {\displaystyle S} is uncountable, S {\displaystyle S} must break up into uncountably many orbits under the action of G {\displaystyle
Axiom_of_choice
Tree in set theory
In set theory, an Aronszajn tree is a tree of uncountable height with no uncountable branches and no uncountable levels. For example, every Suslin tree
Aronszajn_tree
Axiom of set theory
axiom of foundation) is an axiom of Zermelo–Fraenkel set theory that states that every non-empty set A contains an element that is disjoint from A. In first-order
Axiom_of_regularity
UNCOUNTABLE SET
UNCOUNTABLE SET
Surname or Lastname
English
English : patronymic from Setter.
Surname or Lastname
English
English : habitational name from a place in North Yorkshire, so named from Old English setl ‘seat’, ‘dwelling’.
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
Boy/Male
Assamese, Celebrity, Gujarati, Hindu, Indian, Jain, Kannada, Malayalam, Marathi, Oriya, Punjabi, Sanskrit, Sikh, Tamil, Telugu, Traditional
Sky; Lord of Day; Uncountable; Space
Surname or Lastname
English
English : habitational name from a place in Shropshire, so named from Welsh mynydd ‘hill’ + Old English tūn ‘enclosure’, ‘settlement’.
Male
Greek
(Σήθι) Greek form of Egyptian Seti, SETHI means "of Seth."Â
Surname or Lastname
English
English : topographic name for someone who lived in the center of a village, from Middle English midde ‘mid’ + toun ‘village’, ‘town’.English : habitational name from places in Lancashire, Worcestershire, and West Yorkshire, so named in Old English as ‘farmstead at a river confluence’, from (ge)m̄ðe ‘river confluence’ + tūn ‘farmstead’, ‘settlement’.
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.
Male
Hindi/Indian
(सेठ) Hindi name derived from the Sanskrit word setu, SETH means "bridge." Compare with other forms of Seth.
Boy/Male
Hindu, Indian
Uncountable
Boy/Male
Hindu, Indian
Un Countable; Multiple; Countless
Boy/Male
Gujarati, Hindu, Indian, Kannada, Malayalam, Marathi, Oriya, Punjabi, Sikh, Sindhi, Tamil, Telugu
Brave; Winner; Smart; Strong; Uncountable; Infinite God
Boy/Male
Hindu, Indian
Uncountable
Male
Italian
Italian form of Roman Latin Septimus, SETTIMIO means "seventh."
Female
Japanese
(節å) Japanese name SETSUKO means "temperate child."
Surname or Lastname
English
English : habitational name from Mitcham in Surrey, so named from Old English micel ‘big’ + hÄm ‘homestead’, ‘settlement’.
Surname or Lastname
English
English : habitational name from Milwich in Staffordshire, so named from Old English myln ‘mill’ + wīc ‘dairy farm’; ‘(trading) settlement’.
Surname or Lastname
English and Scottish
English and Scottish : habitational name from any of the numerous and widespread places so called. The majority of these are named with Old English middel ‘middle’ + tūn ‘enclosure’, ‘settlement’; a smaller group, with examples in Cumbria, Kent, Northamptonshire, Northumbria, Nottinghamshire, and Staffordshire, have as their first element Old English mylen ‘mill’.
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 : habitational name from places in Cheshire and East Yorkshire, so named from Old English mylen ‘mill’ + tūn ‘enclosure’, ‘settlement’.
UNCOUNTABLE SET
UNCOUNTABLE SET
Girl/Female
Biblical
Separation, division.
Girl/Female
Indian, Telugu
Small Utensil
Boy/Male
Scottish
From the winding valley.
Boy/Male
Hindu, Indian, Kurdish, Telugu
Winner; Happy
Girl/Female
Hindu
The Goddess who is vedic in form
Girl/Female
Indian, Sikh
Helping
Boy/Male
British, English
Manor-friend
Boy/Male
Indian
Light of God
Girl/Female
Tamil
Kajalsri | காஜாலà¯à®¸à®°à¯€Â
Eye liner
Boy/Male
Hindu, Indian
Raft; Heaven
UNCOUNTABLE SET
UNCOUNTABLE SET
UNCOUNTABLE SET
UNCOUNTABLE SET
UNCOUNTABLE SET
a.
Incapable of being daunted; intrepid; fearless; indomitable.
a.
Capable of being accounted for; explicable.
a.
Not to be accounted for; inexplicable; not consonant with reason or rule; strange; mysterious.
n.
One who renders account; one accountable.
n.
The state or quality of being numerable or countable.
n.
The state of being accountable; liability to be called on to render an account; accountableness.
n.
The quality or state of being unaccountable.
a.
Not accountable or responsible; free from control.
v. t.
To submit; to make accountable.
a.
Indubitable.
n.
The quality or state of being accountable; accountability.
n.
The state of being responsible, accountable, or answerable, as for a trust, debt, or obligation.
a.
Accountable.
a.
Liable to be brought to account or punishment; answerable; responsible; accountable; as, amenable to law.
a.
Liable to be called on to render an account; answerable; as, every man is accountable to God for his conduct.
adv.
In an accountable manner.
n.
That for which anyone is responsible or accountable; as, the resonsibilities of power.
a.
See Accountable.
v. t.
Accountable; responsible; sensitive.
a.
Capable of being numbered.