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Function that returns cardinal numbers
a cardinal function (or cardinal invariant) is a function that returns cardinal numbers. The most frequently used cardinal function is the function that
Cardinal_function
Special mathematical function defined as sin(x)/x
zeroth-order spherical Bessel function of the first kind. The sinc function is also called the cardinal sine function. The sinc function has two forms, normalized
Sinc_function
Size of a possibly infinite set
or # A . {\displaystyle \#A.} Cardinality is defined in terms of bijective functions. Two sets have the same cardinality if, and only if, there is a one-to-one
Cardinal_number
Size of a set in mathematics
definition of the cardinality function, by assigning each set to its equinumerous aleph. Basic arithmetic can be done on cardinal numbers in a very natural
Cardinality
Mathematical function such that every output has at least one input
surjective function (also known as surjection, or onto function /ˈɒn.tuː/) is a function f such that, for every element y of the function's codomain, there
Surjective_function
Infinite cardinal number
number Gimel function Regular cardinal Infinity Transfinite number Ordinal number Given the axiom of choice, every infinite set has a cardinality that is an
Aleph_number
In contrast with ordinal utility, in economics
an early conception of cardinality. Bernoulli's imaginary logarithmic utility function and Gabriel Cramer's U = W1/2 function were conceived at the time
Cardinal_utility
Function that preserves distinctness
In mathematics, an injective function (also known as injection, or one-to-one function) is a function f that maps distinct elements of its domain to distinct
Injective_function
Number of arguments required by a function
Parameter p-adic number Cardinality Valency (linguistics) n-ary code n-ary group Function prototype – Declaration of a function's name and type signature
Arity
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
Theorem in axiomatic set theory
In axiomatic set theory, the gimel function is the following function mapping cardinal numbers to cardinal numbers: ℷ : κ ↦ κ c f ( κ ) {\displaystyle
Gimel_function
One-to-one correspondence
two sets are said to have the same cardinal number if there exists a bijection between them. A bijective function from a set to itself is also called
Bijection
Mathematical function that can be computed by a program
Computable functions are the basic objects of study in computability theory. Informally, a function is computable if there is an algorithm that computes
Computable_function
Energy contained within a system
potentials and Massieu functions. The entropy as a function only of extensive state variables is the one and only cardinal function of state for the generation
Internal_energy
Type of cardinal spline
( t ) {\displaystyle \mathbf {F} (t)} . The blending functions are following cardinal functions: C 0 , k ( t ) = ∑ i = 0 k [ ∏ j = i − k j ≠ 0 i ( t j
Catmull–Rom_spline
Collection of sets in mathematics that can be defined based on a property of its members
universal class), the class of all ordinal numbers, and the class of all cardinal numbers. One way to prove that a class is proper is to place it in bijection
Class_(set_theory)
Type of infinite number in set theory
set theory, a cardinal number is a strongly inaccessible cardinal if it is uncountable, regular, and a strong limit cardinal. A cardinal is a weakly inaccessible
Inaccessible_cardinal
Input to a mathematical function
of a function is a value provided to obtain the function's result. It is also called an independent variable. For example, the binary function f ( x
Argument_of_a_function
Function that ranks states of society according to their desirability
voting) functions only use ordinal information; i.e., whether one choice is better than another. Cardinal (or rated voting) functions also use cardinal information;
Social_welfare_function
Function returning one of only two values
switching function, used especially in older computer science literature, and truth function (or logical function), used in logic. Boolean functions are the
Boolean_function
Set theory concept
field of set theory, a large cardinal property is a certain kind of property of transfinite cardinal numbers. Cardinals with such properties are, as the
Large_cardinal
a cardinal number, the cardinal function yields the cardinality of the power set of a set of the given cardinality. Continuum hypothesis Cardinality of
Continuum_function
Collection of mathematical objects
cardinality, a bijection being provided by the function x ↦ tan ( π x / 2 ) {\displaystyle x\mapsto \tan(\pi x/2)} . Having the same cardinality
Set_(mathematics)
Type of cardinal number in mathematics
cardinal is a cardinal number that is equal to its own cofinality. More explicitly, this means that κ {\displaystyle \kappa } is a regular cardinal if
Regular_cardinal
inaccessible cardinals Existence of Mahlo cardinals Existence of measurable cardinals (first conjectured by Ulam) Existence of supercompact cardinals The following
List of statements independent of ZFC
List_of_statements_independent_of_ZFC
Mathematical set containing no elements
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 ensure
Empty_set
Infinite set that is not countable
That is, X is nonempty and there is no surjective function from the natural numbers to X. The cardinality of X is neither finite nor equal to ℵ 0 {\displaystyle
Uncountable_set
Target set of a mathematical function
mathematics, a codomain or set of destination of a function is a set into which all of the outputs of the function are constrained to fall. It is the set Y in
Codomain
Method of deriving conclusions
inherent in logical operators found in statements, making the meaning and function of these operators explicit without adding any additional information.
Rule_of_inference
Mathematical-logic system based on functions
the function space D → D, of functions on itself. However, no nontrivial such D can exist, by cardinality constraints because the set of all functions from
Lambda_calculus
Senior church official
A cardinal is a senior member of the clergy of the Catholic Church. As titular members of the clergy of the Diocese of Rome, they serve as advisors to
Cardinal_(Catholic_Church)
Subset of a function's codomain
a function may refer either to the codomain of the function, or the image of the function. In some cases the codomain and the image of a function are
Range_of_a_function
Mathematical set of all subsets of a set
demonstrated below. An indicator function or a characteristic function of a subset A of a set S with the cardinality |S| = n is a function from S to the two-element
Power_set
Axiom of set theory
proved in ZFC itself, but requires a mild large cardinal assumption (the existence of an inaccessible cardinal). The much stronger axiom of determinacy, or
Axiom_of_choice
Branch of mathematics that studies sets
transfinite numbers, called cardinals and ordinals, which extended the arithmetic of the natural numbers. His notation for the cardinal numbers was the Hebrew
Set_theory
Process of repeating items in a self-similar way
where a function being defined is applied within its own definition. While this apparently defines an infinite number of instances (function values),
Recursion
Function uniquely mapping two numbers into a single number
that integers and rational numbers have the same cardinality as natural numbers. A pairing function is a bijection π : N × N → N . {\displaystyle \pi
Pairing_function
3-volume treatise on mathematics, 1910–1913
than functions, and is quite similar to the type system of PM.) In PM, cardinals are defined as classes of similar classes, whereas in ZFC cardinals are
Principia_Mathematica
Ordered listing of items in collection
initial segment {1, ..., n} of the natural numbers, in which case, its cardinality is n. The empty set is finite, as it can be enumerated by means of the
Enumeration
Standard system of axiomatic set theory
of certain sets and cardinal numbers whose existence was taken for granted by most set theorists of the time, notably the cardinal number aleph-omega (
Zermelo–Fraenkel_set_theory
encountered mathematical objects include numbers, expressions, shapes, functions, and sets. Mathematical objects can be very complex; for example, theorems
Mathematical_object
Proof in set theory
here possible as well. So the cardinal relation fails to be antisymmetric. Consequently, also in the presence of function space sets that are even classically
Cantor's_diagonal_argument
Existence and cardinality of models of logical theories
countable if the set of function and relation symbols in it is countable, and in general the cardinality of a signature is the cardinality of the set of all
Löwenheim–Skolem_theorem
Mathematical set formed from two given sets
_{i\in I}X} is the set of all functions from I to X, and is frequently denoted XI. This case is important in the study of cardinal exponentiation. An important
Cartesian_product
Paradox in set theory
the function F(fx) could be its own argument: in that case there would be a proposition F(F(fx)), in which the outer function F and the inner function F
Russell's_paradox
Mathematical concept
if, for every function f: [κ]<ω → {0, 1} there is a set A of cardinality κ that is homogeneous for f. That is, for every n, the function f is constant
Ramsey_cardinal
Limitative results in mathematical logic
but ZFC + "there exists an inaccessible cardinal" proves ZFC is consistent because if κ is the least such cardinal, then Vκ sitting inside the von Neumann
Gödel's incompleteness theorems
Gödel's_incompleteness_theorems
Set whose elements all belong to another set
the subset and the whole set are infinite, and the subset has the same cardinality (the concept that corresponds to size, that is, the number of elements
Subset
Problem in computer science
often in discussions of computability since it demonstrates that some functions are mathematically definable but not computable. A key part of the formal
Halting_problem
Function, homomorphism, or morphism
In mathematics, a map or mapping is a function in its general sense.[vague] These terms may have originated as from the process of making a geographical
Map_(mathematics)
In mathematics, a solution to a modified form of the confluent hypergeometric equation
1016/0024-3795(95)00705-9. ISSN 0024-3795. Whittaker, J. M. (May 1927). "On the Cardinal Function of Interpolation Theory". Proceedings of the Edinburgh Mathematical
Whittaker_function
Number
empty set is the von Neumann cardinal assignment for a set with no elements, which is the empty set. The cardinality function, applied to the empty set,
0
Statement that is taken to be true
as Morse–Kelley set theory or set theory with a strongly inaccessible cardinal allowing the use of a Grothendieck universe is used, but in fact, most
Axiom
Symbol representing a property or relation in logic
predicates are understood to be characteristic functions or set indicator functions (i.e., functions from a set element to a truth value). Set-builder
Predicate_(logic)
Yes/no problem in computer science
function problem can be turned into a decision problem; the decision problem is just the graph of the associated function. (The graph of a function f
Decision_problem
Theorem for proving more complex theorems
Often, a theorem is broken into multiple cases (for example, a quadratic function may have no real roots, one double root, or two distinct roots), and each
Lemma_(mathematics)
Impossible task in computing
that the intuitive notion of "effectively calculable" is captured by the functions computable by a Turing machine (or equivalently, by those expressible
Entscheidungsproblem
Every set is smaller than its power set
consequence, the cardinality of the real numbers, which is the same as that of the power set of the integers, is strictly larger than the cardinality of the integers;
Cantor's_theorem
Complexity class used to classify decision problems
and PH ⊆ BPP. NP is a class of decision problems; the analogous class of function problems is FNP. The only known strict inclusions come from the time hierarchy
NP_(complexity)
Basic framework of mathematics
involved new methods of reasoning and new basic concepts (continuous functions, derivatives, limits) that were not well founded, but had astonishing
Foundations_of_mathematics
Symbol representing a mathematical concept
systems particularly mathematical logic, a function symbol is a non-logical symbol which represents a function or mapping on the domain of discourse, though
Function_symbol
Mathematical set that can be enumerated
numbers. Equivalently, a set is countable if there exists an injective function from it into the natural numbers; this means that each element in the set
Countable_set
Set of the elements not in a given subset
cardinality Function/Map domain codomain image In/Sur/Bi-jection Schröder–Bernstein theorem Isomorphism Gödel numbering Enumeration Large cardinal inaccessible
Complement_(set_theory)
Algebraic manipulation of "true" and "false"
complement function, the dual function and the contradual function (complemented dual). These four functions form a group under function composition
Boolean_algebra
Value indicating the relation of a proposition to truth
Boolean domain. Corresponding semantics of logical connectives are truth functions, whose values are expressed in the form of truth tables. Logical biconditional
Truth_value
Mathematical concept
r_{\alpha }\mid \alpha <\beta \rangle } , where β is an ordinal with the cardinality of the continuum. Let v0 equal r0. Then let v1 equal rα1, where α1 is
Transfinite_induction
Mathematical operation with two operands
arity two. More specifically, a binary operation on a set is a binary function that maps every pair of elements of the set to an element of the set. Examples
Binary_operation
Topics referred to by the same term
polynomial sequence arising in rational trigonometry Spread (topology), a cardinal function defined on topological spaces, also known as the hereditary cellularity
Spread
Additional mathematical object
preserve algebraic structures; continuous functions, which preserve topological structures; and differentiable functions, which preserve differential structures
Mathematical_structure
Theorem in set theory
injective functions f : A → B and g : B → A between the sets A and B, then there exists a bijective function h : A → B. In terms of the cardinality of the
Schröder–Bernstein_theorem
Proposition in mathematical logic
the possible sizes of infinite sets. It states: There is no set whose cardinality is strictly between that of the integers and the real numbers. The name
Continuum_hypothesis
In logic, a statement which is always true
be deduced from the truth or falsity of each variable. A valuation is a function that assigns each propositional variable to either T (for truth) or F (for
Tautology_(logic)
Set theory concept
explicitly after stage 5. The set Vω has the same cardinality as ω. The set Vω+1 has the same cardinality as the set of real numbers. In the standard Zermelo–Fraenkel
Von_Neumann_universe
Concept in economics and decision theory
transitions between two bundles of goods. A cardinal utility function can be transformed to another utility function by a positive linear transformation (multiplying
Utility
Mathematical use of "for all"
found in the Quantifier article. The negation of a universally quantified function is obtained by changing the universal quantifier into an existential quantifier
Universal_quantification
Area of mathematical logic
introduced to classify theories by their numbers of models in a given cardinality, stability theory proved crucial to understanding the geometry of definable
Model_theory
Yes-or-no question that cannot ever be solved by a computer
answer. Such a problem is said to be undecidable if there is no computable function that correctly answers every question in the problem set. The connection
Undecidable_problem
Finite collection of distinct objects
this equivalence. Any injective function between two finite sets of the same cardinality is also a surjective function (a surjection). Similarly, any surjection
Finite_set
Type of logical system
discourse (over which the quantified variables range), finitely many functions from that domain to itself, finitely many predicates defined on that domain
First-order_logic
Intersection of Set Theory and General Topology
Moore space question was eventually proved to be independent of ZFC. Cardinal functions are widely used in topology as a tool for describing various topological
Set-theoretic_topology
Any one of the distinct objects that make up a set in set theory
known as cardinality; informally, this is the size of a set. In the above examples, the cardinality of the set A is 4, while the cardinality of set B
Element_of_a_set
Subfield of mathematics
Whitehead was published. This seminal work developed the theory of functions and cardinality in a completely formal framework of type theory, which Russell
Mathematical_logic
Concept in logic
cardinality Function/Map domain codomain image In/Sur/Bi-jection Schröder–Bernstein theorem Isomorphism Gödel numbering Enumeration Large cardinal inaccessible
Logical_equivalence
Function computable with bounded loops
In computability theory, a primitive recursive function is, roughly speaking, a function that can be computed by a computer program whose loops are all
Primitive_recursive_function
Set that is not a finite set
if and only if for every natural number, the set has a subset whose cardinality is that natural number. If the axiom of choice holds, then a set is infinite
Infinite_set
Argument whose conclusion must be true if its premises are
classical logic. However, within that system 'true' and 'false' essentially function more like mathematical states such as binary 1s and 0s than the philosophical
Validity_(logic)
Syntactically correct logical formula
constant symbols, predicate symbols, and function symbols of the theory at hand, along with the arities of the function and predicate symbols. The definition
Well-formed_formula
Computation model defining an abstract machine
\rightharpoonup Q\times \Gamma \times \{L,R\}} is a partial function called the transition function, where L is left shift, R is right shift. If δ {\displaystyle
Turing_machine
Fundamental theorem in mathematical logic
However the situation is different when the language is of arbitrary large cardinality since then, though the completeness and compactness theorems remain provably
Gödel's_completeness_theorem
Mathematical set containing all objects
set of any set (whether infinite or not) always has strictly higher cardinality than the set itself. The difficulties associated with a universal set
Universal_set
Set of elements in any of some sets
Retrieved 2025-04-10. Pierpont, James (1912). Lectures On The Theory Of Functions Of Real Variables Vol II. Osmania University, Digital Library Of India
Union_(set_theory)
Mathematical use of "there exists"
union of sets. A quantified propositional function is a statement; thus, like statements, quantified functions can be negated. The ¬ {\displaystyle \lnot
Existential_quantification
Cardinality of the set of real numbers
Cantor defined cardinality in terms of bijective functions: two sets have the same cardinality if, and only if, there exists a bijective function between them
Cardinality_of_the_continuum
Set of elements common to all of some sets
\tau } ). Algebra of sets – Identities and relationships involving sets Cardinality – Size of a set in mathematics Complement – Set of the elements not in
Intersection_(set_theory)
Relationship where one statement follows from another
algebraic logic Ampheck Boolean algebra (logic) Boolean domain Boolean function Boolean logic Causality Deductive reasoning Logic gate Logical graph Peirce's
Logical_consequence
Mathematical proposition equivalent to the axiom of choice
the successor cardinal (= least ordinal whose cardinality is larger than κ {\displaystyle \kappa } ). Then, for the reason of cardinality, κ + ↪̸ P , β
Zorn's_lemma
Pair of mathematical objects
formalized set theories and is methodologically similar to defining the cardinal of a set as the class of all sets equipotent with the given set. Morse–Kelley
Ordered_pair
Thesis on the nature of computability
Church–Turing thesis is a thesis about the nature of computable functions. It states that a function on the natural numbers can be calculated by an effective
Church–Turing_thesis
Mathematical property of a space
. {\displaystyle P.} The cardinality | X | {\displaystyle \vert X\vert } of the space X {\displaystyle X} . The cardinality | τ ( X ) | {\displaystyle
Topological_property
All-encompassing set or class
Cantor was using when he first developed modern naive set theory and cardinality in the 1870s and 1880s in applications to real analysis. The only sets
Universe_(mathematics)
CARDINAL FUNCTION
CARDINAL FUNCTION
Girl/Female
American, Christian, Finnish, French, Indian, Italian, Latin, Swedish, Tamil
Beloved; Keel of a Ship; Pure; Dear Little One; Darling
Boy/Male
Shakespearean
King Richard III' Cardinal Bourchier, Archbishop of Canterbury.
Girl/Female
Arabic, Farsi, Indian
Justified Love; Love; Decorated; Justified
Boy/Male
Sikh
One on whom there is gods grace, Gods mercy
Girl/Female
Australian, Latin
Little Darling
Girl/Female
American, British, English, Hebrew, Latin, Lebanese, Spanish
Song; Garden; Orchard; Vineyard
Girl/Female
English Spanish
Song.
Female
English
 19th-century English elaborated form of Latin cara, CARINA means "beloved." From the constellation Carina, from Latin carina, which originally meant "shell of a nut," later "keel of a ship."
Girl/Female
Biblical
Carnal, fleshly.
Surname or Lastname
English, French, Spanish, and Dutch
English, French, Spanish, and Dutch : from Middle English, Old French cardinal ‘cardinal’, the church dignitary (Latin cardinalis, originally an adjective meaning ‘crucial’). The surname may have denoted a servant who worked in a cardinal’s household, but was probably more often bestowed as a nickname on someone who habitually dressed in red or who had played the part of a cardinal in a pageant, or on one who acted in a lordly and patronizing manner, like a prince of the Church.A bearer of the name, of unknown origin, is documented in Montreal by 1666.
Boy/Male
Shakespearean
King John' Cardinal Pandulph, the Pope's legate.
Girl/Female
Latin
Ardent. Eager. Industrious.
Biblical
carnal; fleshly
Girl/Female
French Swedish American Italian Latin
Pure.
Surname or Lastname
English
English : variant spelling of Carnell.French : metonymic occupational name for a maker of latches and hinges, from Old Picard carnel, Old French charnel ‘hinge’.
Surname or Lastname
English
English : variant of Cordell.
Boy/Male
Christian, French, Greek, Indian, Latin
Carnal Love
Girl/Female
Australian, British, Danish, English, German
Female Version of Carl
Boy/Male
Shakespearean
King Henry the Eighth' Cardinal Campeius.
Boy/Male
Hindu, Indian, Punjabi, Sikh
One on whom There is God's Grace
CARDINAL FUNCTION
CARDINAL FUNCTION
Female
Finnish
Pet form of Finnish Piritta, PIRJO means "exalted one."
Boy/Male
Bengali, Indian, Modern
Disaster
Boy/Male
Hindu, Indian, Tamil, Telugu
Sweet; Humble; Polite; Knowledgeable; Modest
Boy/Male
Arabic, Muslim
The Righteousness of the Faith
Boy/Male
Tamil
Ruler
Girl/Female
Hindu, Indian
Queen of Nights
Boy/Male
Indian, Tamil
Lord Vishnu
Female
Hebrew
(×ֶמֶת) Hebrew name EMET means "truth." The masculine form is spelled Emmet.
Boy/Male
Arabic, Muslim
Proud
Boy/Male
Indian, Tamil
Fate
CARDINAL FUNCTION
CARDINAL FUNCTION
CARDINAL FUNCTION
CARDINAL FUNCTION
CARDINAL FUNCTION
a.
Of fundamental importance; preeminent; superior; chief; principal.
v. t.
To depose from the rank of cardinal.
a.
The act or process of preparing staple for spinning, etc., by carding it. See the Note under Card, v. t.
v. t.
A roll of wool or other fiber as it comes from the carding machine.
a.
A woman's short cloak with a hood.
a.
Exciting action in the heart, through the medium of the stomach; cordial; stimulant.
a.
Pertaining to, resembling, or hear the heart; as, the cardiac arteries; the cardiac, or left, end of the stomach.
a.
Cardiac.
n.
Any invigorating and stimulating preparation; as, a peppermint cordial.
n.
The anterior or cardiac orifice of the stomach, where the esophagus enters it.
n.
A medicine which excites action in the stomach; a cardial.
v. t.
To exalt to the office of a cardinal.
a.
One of the ecclesiastical princes who constitute the pope's council, or the sacred college.
a.
Written or printed in the margin; as, a marginal note or gloss.
n.
The office, rank, or dignity of a cardinal.
a.
Mulled red wine.
a.
Indicating order or succession; as, the ordinal numbers, first, second, third, etc.
n.
The condition, dignity, of office of a cardinal
n.
The cardinal bird.
n.
A cardinalate. See Cardinal.