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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
the set) is not greater than that of the natural numbers. A countable set that is not finite is said to be countably infinite; for example the set of all
Countable_set
Mathematical concept
infinity by studying infinite sets and infinite numbers, showing that they can be of various sizes. For example, if a line is viewed as the set of all of its
Infinity
Set with an equinumerous proper subset
In mathematics, a set A is Dedekind-infinite (named after the German mathematician Richard Dedekind) if some proper subset B of A is equinumerous to A
Dedekind-infinite_set
Generalization of "n-th" to infinite cases
set theory, an ordinal number, or ordinal, is a generalization of ordinal numerals (first, second, nth, etc.) aimed to extend enumeration to infinite
Ordinal_number
Collection of mathematical objects
process—and were reluctant to consider infinite sets.[citation needed] For example, a line was considered not as a set of points, but as a locus where a point
Set_(mathematics)
Set of elements in any of some sets
B = {1, 2, 3, 4, 5, 6, 7}. A more elaborate example (involving two infinite sets) is: A = {x is an even integer greater than 1} B = {x is an odd integer
Union_(set_theory)
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
Thought experiment of infinite sets
(colloquially the Infinite Hotel Paradox or Hilbert's Hotel) is a thought experiment which illustrates a counterintuitive property of infinite sets. It shows
Hilbert's paradox of the Grand Hotel
Hilbert's_paradox_of_the_Grand_Hotel
Mathematician (1845–1918)
one-to-one correspondence between the members of two sets, defined infinite and well-ordered sets, and proved that the real numbers are more numerous than
Georg_Cantor
Set of points on a line segment with certain topological properties
Cantor ternary set contains all points in the interval [ 0 , 1 ] {\displaystyle [0,1]} that are not deleted at any step in this infinite process. The same
Cantor_set
Standard system of axiomatic set theory
} where Z 0 {\displaystyle Z_{0}} is any infinite set and P {\displaystyle {\mathcal {P}}} is the power set operation. Moreover, one of Zermelo's axioms
Zermelo–Fraenkel_set_theory
contradictions within modern axiomatic set theory. Set theory as conceived by Georg Cantor assumes the existence of infinite sets. As this assumption cannot be
Paradoxes_of_set_theory
Axiom of Zermelo-Fraenkel set theory
of at least one infinite set, namely a set containing the natural numbers. It was first published by Ernst Zermelo as part of his set theory in 1908.
Axiom_of_infinity
Branch of mathematics that studies sets
of sets to mathematics. In his work, he (among other things) expanded on Galileo's paradox, and introduced one-to-one correspondence of infinite sets, for
Set_theory
Any one of the distinct objects that make up a set in set theory
cardinality of set B and set C are both 3. An infinite set is a set with an infinite number of elements, while a finite set is a set with a finite number
Element_of_a_set
Infinite cardinal number
particularly in set theory, the aleph numbers are a sequence of numbers used to represent the cardinality (or size) of infinite sets. They were introduced
Aleph_number
Basic framework of mathematics
to systematically study infinite sets. In particular, he introduced cardinal numbers that measure the size of infinite sets, and ordinal numbers that
Foundations_of_mathematics
Number that is larger than all finite numbers
quantify the size of infinite sets, and the transfinite ordinals, which are ordinal numbers used to provide an ordering of infinite sets. The term transfinite
Transfinite_number
Size of a set in mathematics
cardinality is an inherent property of sets, roughly meaning the number of individual objects they contain, which may be infinite. The concept is understood through
Cardinality
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
Power_set
Counterintuitive result in probability
The infinite monkey theorem states that a monkey hitting keys independently and at random on a typewriter keyboard for an infinite amount of time will
Infinite_monkey_theorem
Set of natural numbers
mathematics, an IP set is a set of natural numbers which contains all finite sums of some infinite set. The finite sums of a set D of natural numbers
IP_set
Concept in the philosophy of mathematics
the natural numbers form a set (necessarily infinite). A great discovery of Cantor is that, if one accepts infinite sets, then there are different sizes
Actual_and_potential_infinity
Infinite set not splittable into infinite sets
In set theory, an amorphous set is an infinite set that is not the disjoint union of two infinite subsets. Amorphous sets cannot exist if the axiom of
Amorphous_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
Size of a possibly infinite set
The behavior of cardinalities of infinite sets is more complex. For example, there exists a bijection between the set of all natural numbers N {\displaystyle
Cardinal_number
First article on transfinite set theory
Cantor's first set theory article contains Georg Cantor's first theorems of transfinite set theory, which studies infinite sets and their properties. One
Cantor's first set theory article
Cantor's_first_set_theory_article
Mathematical set containing no elements
infinity, which guarantees the existence of at least one infinite set, can be used to construct the set of natural numbers, N 0 {\displaystyle \mathbb {N} _{0}}
Empty_set
Concept relating to infinite sets
The numerosity of an infinite set, as initially introduced by the Italian mathematician Vieri Benci and later on extended with the help of Mauro Di Nasso
Numerosity_(mathematics)
Theorem in combinatorics
straightforward application is to finite sets (such as pigeons and boxes), it is also used with infinite sets that cannot be put into one-to-one correspondence
Pigeonhole_principle
Topics referred to by the same term
Look up infinite in Wiktionary, the free dictionary. Infinite may refer to: Infinite set, a set that is not a finite set Infinity, an abstract concept
Infinite
Probability saying
distinction becomes important when the sample space is an infinite set, because an infinite set can have non-empty subsets of probability 0. Some examples
Almost_surely
Finding the number of elements of a finite set
to uniquely identifying the elements of a finite (combinatorial) set or infinite set by assigning a number to each element. Counting sometimes involves
Counting
Type of mathematical group
two-dimensional geometry, the infinite dihedral group represents the frieze group symmetry, p1m1, seen as an infinite set of parallel reflections along
Infinite_dihedral_group
Maximal proper filter
are the free ultrafilters. The existence of free ultrafilters on any infinite set is implied by the ultrafilter lemma, which can be proven in ZFC. On the
Ultrafilter_on_a_set
Proof in set theory
are infinite sets which cannot be put into one-to-one correspondence with the infinite set of natural numbers – informally, that there are sets which
Cantor's_diagonal_argument
2021 video game
Halo Infinite is a 2021 first-person shooter game developed by 343 Industries and published by Xbox Game Studios. It is the sixth mainline installment
Halo_Infinite
1996 novel by David Foster Wallace
Infinite Jest is a 1996 novel by American writer David Foster Wallace. Often considered Wallace’s magnum opus, the book has been categorized as an encyclopedic
Infinite_Jest
Dedekind-infinite sets. In ZF, it can be proved that all Dedekind-infinite sets are simply infinite, but the converse – that all simply infinite sets are Dedekind-infinite
Glossary_of_set_theory
Extension of ideas in combinatorics to infinite sets
infinitary combinatorics, or combinatorial set theory, is an extension of ideas in combinatorics to infinite sets. Some of the things studied include continuous
Infinitary_combinatorics
Type of group in abstract algebra
! {\displaystyle n!} . Although symmetric groups can be defined on infinite sets, this article focuses on the finite symmetric groups: their applications
Symmetric_group
2021 film by Antoine Fuqua
Infinite is a 2021 American science fiction action film directed by Antoine Fuqua, from a screenplay written by Ian Shorr based on a story by Todd Stein
Infinite_(film)
Process of repeating items in a self-similar way
apparently defines an infinite number of instances (function values), it is often done in such a way that no infinite loop or infinite chain of references
Recursion
German mathematician (1831–1916)
invoked similarity to give the first precise definition of an infinite set: a set is infinite when it is "similar to a proper part of itself," in modern
Richard_Dedekind
Order whose elements are all comparable
Weyer, Mark (2002). "Decidability of S1S and S2S". Automata, Logics, and Infinite Games. Lecture Notes in Computer Science. Vol. 2500. Springer. pp. 207–230
Total_order
Axiom of set theory
each set, even if the collection is infinite. Formally, the axiom establishes existence rather than a construction; it states that for every set I {\displaystyle
Axiom_of_choice
Theorem equivalent to the Axiom of Choice
that in ZF the statement "For every infinite set A {\displaystyle A} , there is a bijective map between the sets A {\displaystyle A} and A × A {\displaystyle
Tarski's_theorem_about_choice
Proposition in mathematical logic
specifically set theory, the continuum hypothesis (abbreviated CH) is a hypothesis about the possible sizes of infinite sets. It states: There is no set whose
Continuum_hypothesis
Every set is smaller than its power set
discovery of an argument that is applicable to any set, and shows that the theorem holds for infinite sets also. As a consequence, the cardinality of the
Cantor's_theorem
Concept in philosophy and set theory
papers developing his theory of cardinality (Mächtigkeit, magnitude) of infinite sets. In his first paper, On a Property of the Collection of All Real Algebraic
Absolute_infinite
System of formal deduction in logic
An axiom schema is an infinite set of axioms obtained by substituting all formulas of some form into a specific pattern. The set of logical axioms includes
Hilbert_system
Class of mathematical orderings
does not occur in finite sets, and may or may not occur in an infinite set; the infinite sets without limit point are the sets of order type ω, for example
Well-order
Japanese-American anime streaming television series
Evil: Infinite Darkness (stylized as RESIDENT EVIL: Infinite Darkness) is a Japanese horror-action biopunk CGI original net animation miniseries set in the
Resident Evil: Infinite Darkness
Resident_Evil:_Infinite_Darkness
Marked objects for finding random numbers
of the infinite set of prisms, with triangle faces: any multiple of 4 (so that a facet faces up), starting from 8 Disphenoids, an infinite set of tetrahedra
Dice
Paradox in set theory
Galileo's paradox is a demonstration of one of the surprising properties of infinite sets. In his final scientific work, Two New Sciences, Galileo Galilei made
Galileo's_paradox
Posthumous 1851 treatise by Bernard Bolzano on mathematical infinity
prehistory of set theory for its detailed defence of actual (completed) infinity and its analysis of one-to-one pairings between infinite “multitudes”
Paradoxes_of_the_Infinite
System of mathematical set theory
GST is sufficient for all mathematics not requiring infinite sets, and is the weakest known set theory whose theorems include the first-order Peano axioms
General_set_theory
Yes-or-no question that cannot ever be solved by a computer
run. A decision problem is a question which, for every input in some infinite set of inputs, requires a "yes" or "no" answer. Those inputs can be numbers
Undecidable_problem
Axiomatic set theories based on the principles of mathematical constructivism
\omega } into it. A set that is even in bijection with ω {\displaystyle \omega } may be called countably infinite. A set is Tarski-infinite if there is a chain
Constructive_set_theory
Unary operation on string sets
{\displaystyle V} is any finite or countably infinite set of characters, then V ∗ {\displaystyle V^{*}} is a countably infinite set. As a result, each formal language
Kleene_star
Philosophy of mathematics that accepts only finite objects
to the mainstream philosophy of mathematics where infinite mathematical objects (e.g., infinite sets) are accepted as existing. The main idea of finitistic
Finitism
Means of constructing a group from two subgroups
sum of 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,
Direct_sum_of_groups
About mathematical infinity
theory of infinite sets was first developed by Georg Cantor. Although this work has become a thoroughly standard fixture of classical set theory, it
Controversy over Cantor's theory
Controversy_over_Cantor's_theory
1960 mathematics textbook by Paul Halmos
\{0\}\}} and so on. This implies that there is a set containing all the elements of the first infinite von Neumann ordinal ω {\displaystyle \omega } .
Naive_Set_Theory_(book)
Jungian theory
within the Self, with Jung viewing parts of the self as part of the infinite set of archetypes within the collective unconscious. Modern Jungian clinical
Anima_and_animus
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
Subset with finite complement
one says the set is cocountable. These arise naturally when generalizing structures on finite sets to infinite sets, particularly on infinite products, as
Cofiniteness
2013 video game
BioShock series, Infinite was released worldwide for the PlayStation 3, Windows, Xbox 360, and OS X platforms in 2013. The game is set in the year 1912
BioShock_Infinite
Paradox in set theory
Edward N. (ed.). The Stanford Encyclopedia of Philosophy. R. Bunn, Infinite Sets and Numbers (1967), pp.176–178. Ph.D dissertation, University of British
Russell's_paradox
Concept in mathematics
here is a proof (from ZF + ACω) that every infinite set is Dedekind-infinite: Let X {\displaystyle X} be infinite. For each natural number n {\displaystyle
Axiom_of_countable_choice
Subset which is both open and closed
way, the components will be clopen. Now let X {\displaystyle X} be an infinite set under the discrete metric – that is, two points p , q ∈ X {\displaystyle
Clopen_set
Philosophical theory
denumerably infinite. This is not, in fact, a theorem of mathematics. But, according to Hartman, people are capable of a denumerably infinite set of predicates
Science_of_value
Limited DC comic crossover series
Crisis on Infinite Earths is a 1985 to 1986 American comic book crossover series published by DC Comics. Written by Marv Wolfman and pencilled by George
Crisis_on_Infinite_Earths
Largest and smallest value taken by a function at a given point
set theory, the maximum and minimum of a set are the greatest and least elements in the set, respectively. Unbounded infinite sets, such as the set of
Maximum_and_minimum
Term in set theory
set theory, when dealing with sets of infinite size, the term almost or nearly is used to refer to all but a negligible amount of elements in the set
Almost
Generalised alphabetical order
least element). It is not true that the set of all finite words is well-ordered; for example, the infinite set of words {b, ab, aab, aaab, ... } has no
Lexicographic_order
Finiteness of sets of forbidden graph minors
infinite antichains, infinite sets of graphs that are all unrelated to each other by the minor ordering. If S {\displaystyle {\mathcal {S}}} is a set
Robertson–Seymour_theorem
Alternative mathematical set theory
Pocket set theory (PST) is an alternative set theory in which there are only two infinite cardinal numbers, ℵ0 (aleph-naught, the cardinality of the set of
Pocket_set_theory
One-to-one correspondence
definition to infinite sets leads to the concept of cardinal number, a way to distinguish the various sizes of infinite sets. Any infinite set that has a
Bijection
Study of discrete mathematical structures
of the term "discrete mathematics". The set of objects studied in discrete mathematics can be finite or infinite. The term finite mathematics is sometimes
Discrete_mathematics
Set of points that satisfy some specified conditions
19th century, mathematicians did not consider infinite sets. Instead of viewing lines and curves as sets of points, they viewed them as places where a
Locus_(mathematics)
Mathematical set formed from two given sets
|A| · |B| · |C| and so on. The set A × B is infinite if either A or B is infinite, and the other set is not the empty set. The Cartesian product can be
Cartesian_product
Form of logic that allows quantification over predicates
(since Cantor's theorem implies that the set of all subsets of a countably infinite set is an uncountably infinite set). This construction is closely related
Second-order_logic
Topology made of cocountable subsets
topology, is a topology that can be defined on any infinite set X {\displaystyle X} . In this topology, a set is open if its complement in X {\displaystyle
Cocountable_topology
Periodic set of points
the real coordinate space R n {\displaystyle \mathbb {R} ^{n}} is an infinite set of points in this space with these properties: Coordinate-wise addition
Lattice_(group)
Result in combinatorics and graph theory
be a finite family of sets (note that although F {\displaystyle {\mathcal {F}}} is not itself allowed to be infinite, the sets in it may be so, and F
Hall's_marriage_theorem
Type of logical system
of set theory or arithmetic. Infinitary logic allows infinitely long sentences. For example, one may allow a conjunction or disjunction of infinitely many
First-order_logic
Decay of strong electromagnetic fields into particles
the infinite set of diagrams shown below, containing one electron loop and any number of external photon legs, each with zero energy. The infinite sum
Schwinger_effect
Axiom(s) of Set Theory
In set theory, several ways have been proposed to construct the natural numbers. These include the representation via von Neumann ordinals, commonly employed
Set-theoretic definition of natural numbers
Set-theoretic_definition_of_natural_numbers
tracial state. Examples of ICC groups are the group of permutations of an infinite set that leave all but a finite subset of elements fixed, and free groups
Infinite conjugacy class property
Infinite_conjugacy_class_property
Set with exactly one element
ultrafilter lemma implies that non-principal ultrafilters exist on every infinite set (these are called free ultrafilters). Every net valued in a singleton
Singleton_(mathematics)
mathematics, semi-infinite objects are objects which are infinite or unbounded in some but not all possible ways. Generally, a semi-infinite set is bounded in
Semi-infinite
System of mathematical set theory
applied to any infinite set of objects whatsoever – there are only so many elements that one can specify in finite time using a finite set of symbols and
Internal_set_theory
Branch of mathematics relating to posets
partially ordered sets. Considering again inclusion-orders of sets, a set is already strictly smaller than another, possibly infinite, set if it contains
Domain_theory
Concept in universal algebra in mathematics
Zbl 0281.08001. Rosenberg, Ivo G. (1976). "The set of maximal closed classes of operations on an infinite set A has cardinality 22|A|". Archiv der Mathematik
Clone_(algebra)
Algebraic manipulation of "true" and "false"
power set 2X of X, consisting of all subsets of X. Here X may be any set: empty, finite, infinite, or even uncountable. Example 2. The empty set and X
Boolean_algebra
Theoretical computer science concept
theoretical computer science, an ω-language is a set of infinite words, where an infinite word is an infinite-length sequence (specifically, an ω-length sequence)
Omega_language
Used to count, measure, and label
the set of all real numbers is uncountably infinite but the set of all algebraic numbers is countably infinite, so there is an uncountably infinite number
Number
Logical principle
finite sets, therefore, Brouwer accepted the principle of the excluded middle as valid. He refused to accept it for infinite sets because if the set S is
Law_of_excluded_middle
INFINITE SET
INFINITE SET
Boy/Male
Tamil
Infinite, Endless
Boy/Male
Tamil
Infinite, Endless
Boy/Male
Hindu
Infinite, Endless
Boy/Male
Tamil
Infinite visionary
Boy/Male
Tamil
Infinite God
Girl/Female
Indian, Telugu
Infinite
Girl/Female
Hindu, Indian, Marathi
Infinite; Matchless
Boy/Male
Hindu, Indian, Marathi, Sanskrit
Infinite
Girl/Female
Tamil
Infinite, Divine
Boy/Male
Hindu
Infinite, Endless
Girl/Female
Indian, Telugu
Infinite
Boy/Male
Japanese
Infinite; endless.
Boy/Male
Indian
Infinite visionary
Boy/Male
Hindi
Infinite.
Girl/Female
Hindi
Infinite.
Girl/Female
Hindu, Indian, Marathi
Infinite; Knowledge
Girl/Female
Hindu, Indian
Infinite
Girl/Female
Indian
Infinite, Divine
Boy/Male
Indian
Infinite.
Boy/Male
Hindu
Infinite God
INFINITE SET
INFINITE SET
Boy/Male
Latin Greek
Name of a king.
Surname or Lastname
English
English : variant spelling of Naylor.
Girl/Female
Indian
Most perfect, Complete
Boy/Male
Arabic, Bengali, Celebrity, Hindu, Indian, Kannada, Muslim, Oriya
The Sun
Boy/Male
Australian, Danish, Dutch, French, German, Greek, Swedish, Welsh
Crowned; Garland; Wreath; Similar to Stephen
Girl/Female
Arabic, Muslim
Sacrifice
Girl/Female
Tamil
Future, Futuristic
Boy/Male
African, Arabic, Hindu, Indian, Malaysian, Muslim, Tamil
Peaceful
Girl/Female
Australian, Vietnamese
Peace
Boy/Male
Hindu, Indian, Sindhi
Take Aashka of Aarti
INFINITE SET
INFINITE SET
INFINITE SET
INFINITE SET
INFINITE SET
n.
Infinite extent; unlimited space; immensity; infinity.
n.
The state or quality of being infinite; infinity; greatness; immensity.
a.
Infinite; perpetual, as a canon whose end leads back to the beginning. See Infinite, a., 5.
n.
The quality or state of being infinite, or without limits; infiniteness.
n.
Unlimited capacity, energy, excellence, or knowledge; as, the infinity of God and his perfections.
n.
That part of a line, or of a plane, or of space, which is infinitely distant. In modern geometry, parallel lines or planes are sometimes treated as lines or planes meeting at infinity.
n.
The Infinite Being; God; the Almighty.
n.
An infinitive form of the verb; a verb in the infinitive mood; the infinitive mood.
a.
Not definite; not limited, defined, or specified; not explicit; not determined or fixed upon; not precise; uncertain; vague; confused; obscure; as, an indefinite time, plan, etc.
n.
An infinity; an incalculable or very great number.
pl.
of Infinity
a.
Having a limit; limited in quantity, degree, or capacity; bounded; -- opposed to infinite; as, finite number; finite existence; a finite being; a finite mind; finite duration.
n.
An infinite quantity or magnitude.
a.
Having certain or distinct; determinate in extent or greatness; limited; fixed; as, definite dimensions; a definite measure; a definite period or interval.
a.
Unlimited or boundless, in time or space; as, infinite duration or distance.
a.
Boundless; infinite.
n.
That which is infinite; boundless space or duration; infinity; boundlessness.
a.
Without limit in power, capacity, knowledge, or excellence; boundless; immeasurably or inconceivably great; perfect; as, the infinite wisdom and goodness of God; -- opposed to finite.
n.
Endless or indefinite number; great multitude; as an infinity of beauties.
a.
Having no determined or certain limits; large and unmeasured, though not infinite; unlimited; as indefinite space; the indefinite extension of a straight line.