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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
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
Mathematical set that can be enumerated
mathematical set is countable if either it is finite or it can be put in one to one correspondence with the set of natural numbers. Equivalently, a set is countable
Countable_set
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
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
the a set is hereditarily P if all elements of its transitive closure have property P. Examples: Hereditarily countable set Hereditarily finite set Hessenberg
Glossary_of_set_theory
Property of objects inherited by all their subobjects
A hereditarily countable set is a countable set of hereditarily countable sets. Assuming the axiom of countable choice, then a set is hereditarily countable
Hereditary_property
inheritance of object-oriented programming. Hereditarily countable set Hereditary property Hierarchy (mathematics) Nested set model for storing hierarchical information
Nested_set_collection
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
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)
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
Type of topological space
Every second-countable space is hereditarily Lindelöf. Every countable space is hereditarily Lindelöf. Every Suslin space is hereditarily Lindelöf. Every
Lindelöf_space
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
Axiomatic set theories based on the principles of mathematical constructivism
{\displaystyle {\mathsf {ZF}}} , this is the set H ℵ 1 {\displaystyle H_{\aleph _{1}}} of hereditarily countable sets and has ordinal rank at most ω 2 {\displaystyle
Constructive_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)
System of mathematical set theory
theorems in set theory, such as the Mostowski collapse lemma. Constructible universe Admissible ordinal Hereditarily countable set Kripke–Platek set theory
Kripke–Platek_set_theory
sets. Another example is the set of hereditarily countable sets. Admissible ordinal Barwise, Jon (1975). Admissible Sets and Structures: An Approach to Definability
Admissible_set
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
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
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)
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
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 theory concept
the empty set considered a special case of an urelement. If ω is the set of natural numbers, then Vω is the set of hereditarily finite sets, which is
Von_Neumann_universe
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
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 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
Hereditarily P A space is hereditarily P for some property P if every subspace is also P. Hereditary A property of spaces is said to be hereditary if
Glossary_of_general_topology
Concept in set theory
In set theory, a code for a hereditarily countable set x ∈ H ℵ 1 {\displaystyle x\in H_{\aleph _{1}}\,} is a set E ⊂ ω × ω {\displaystyle E\subset \omega
Code_(set_theory)
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
System of mathematical set theory
existence of ordinals α ≥ ω + ω, which include uncountably many hereditarily countable sets. This follows from Skolem's result that Vω+ω satisfies Zermelo's
Von Neumann–Bernays–Gödel set theory
Von_Neumann–Bernays–Gödel_set_theory
Alternative mathematical set theory
pocket set theory is given by taking the sets of pocket set theory to be the constructible elements of HC (the set of hereditarily countable sets), and
Pocket_set_theory
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
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)
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
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)
Alternative to the standard Zermelo–Fraenkel set theory
set theory Morse–Kelley set theory Tarski–Grothendieck set theory Ackermann set theory Type theory New Foundations Positive set theory Internal set theory
List of alternative set theories
List_of_alternative_set_theories
System of mathematical set theory
Zermelo set theory (sometimes denoted by Z-), as set out in a seminal paper in 1908 by Ernst Zermelo, is the ancestor of modern Zermelo–Fraenkel set theory
Zermelo_set_theory
Property of topological spaces stronger than normality
called hereditarily collectionwise normal if every subspace of X with the subspace topology is collectionwise normal. In the same way that hereditarily normal
Collectionwise_normal_space
Technical treatment of Boolean algebras
certain hereditarily countable sets. The n-ary Boolean operations themselves constitute a power set algebra 2W, namely when W is taken to be the set of 2n
Boolean algebras canonically defined
Boolean_algebras_canonically_defined
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
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
Mathematical set formed from two given sets
\times \cdots } can be visualized as a vector with countably infinite real number components. This set is frequently denoted R ω {\displaystyle \mathbb
Cartesian_product
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
Topological space which is a generalization of certain compact spaces
space such that every subspace of it is a paracompact space is called hereditarily paracompact. This is equivalent to requiring that every open subspace
Paracompact_space
in V [ G ] {\displaystyle V[G]} , not in V {\displaystyle V} . Hereditarily countable set Schindler, Ralf (2000), "Proper forcing and remarkable cardinals"
Remarkable_cardinal
Class of mathematical set whose elements are all subsets
Neumann, ordinal numbers are defined as hereditarily transitive sets: an ordinal number is a transitive set whose members are also transitive (and thus
Transitive_set
Mathematical set of all subsets of a set
a 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
Power_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 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)
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
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
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
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
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)
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
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
Family of subsets representing "large" sets
Similarly, if X {\displaystyle X} is a set, the cocountable subsets of X {\displaystyle X} (those whose complement is countable) form a filter, the cocountable
Filter_on_a_set
Model of (first-order) Peano arithmetic that contains non-standard numbers
there must exist countable non-standard models of arithmetic. One way to define such a model is to use Henkin semantics. Any countable non-standard model
Non-standard model of arithmetic
Non-standard_model_of_arithmetic
System of mathematical set theory
collection of hereditarily finite sets in M will satisfy the GST axioms. Therefore, GST cannot prove the existence of even a countable infinite set, that is
General_set_theory
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
Function in mathematical logic
than numbers to do the encoding. In simple cases when one uses a hereditarily finite set to encode formulas this is essentially equivalent to the use of
Gödel_numbering
uncountable discrete space; b) the countable product of X with itself is hereditarily separable and hereditarily Lindelöf. Deza, Michel Marie; Deza,
Cosmic_space
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
System of mathematical set theory
example, if MK is consistent then it has a countable first-order model, while second-order ZFC has no countable models. ZFC, NBG, and MK each have models
Morse–Kelley_set_theory
Set with algorithmic membership test
In computability theory, a set of natural numbers is computable (or decidable or recursive) if there is an algorithm that computes the membership of every
Computable_set
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
Topological space that is maximally disconnected
is homeomorphic to a subset of a countable product of discrete spaces. It is in general not true that every open set in a totally disconnected space is
Totally_disconnected_space
Theory that allows sets to be elements of themselves
Non-well-founded set theories (sometimes unhyphenated, as nonwellfounded; or poorly founded) are variants of axiomatic set theory that allow sets to be elements
Non-well-founded_set_theory
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
Pair of logical equivalences
{A_{i}}},\end{aligned}}} where I is some, possibly countably or uncountably infinite, indexing set. In set notation, De Morgan's laws can be remembered using
De_Morgan's_laws
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
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
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
Axiom of set theory
to functions f that can be represented as sets as opposed to undefinable classes. The hereditarily finite sets, Vω, satisfy the axiom of regularity (and
Axiom_of_regularity
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
Concept in axiomatic set theory
power set is one of the Zermelo–Fraenkel axioms of axiomatic set theory. It guarantees for every set x {\displaystyle x} the existence of a set P ( x
Axiom_of_power_set
Concept in set theory
In set theory, a branch of mathematics, an urelement or ur-element (from the German prefix ur-, 'primordial') is an object that is not a set, but that
Urelement
Proof in set theory
proof by contradiction to show that: The set T is uncountable. The proof starts by assuming that T is countable. Then all its elements can be written in
Cantor's_diagonal_argument
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
Kind of transfinite induction
set theory, ∈ {\displaystyle \in } -induction, also called epsilon-induction or set-induction, is a principle that can be used to prove that all sets
Epsilon-induction
Model theory concept
been almost completely solved for the case of a countable theory T. In this section T is a countable complete theory and κ is a cardinal. The Löwenheim–Skolem
Spectrum_of_a_theory
In mathematics, particularly in the subfields of set theory and topology, a set C {\displaystyle C} is said to be saturated with respect to a function
Saturated_set
Mathematical construction of a set with an equivalence relation
setoid (X, ~) is a set (or type) X equipped with an equivalence relation ~. A setoid may also be called E-set, Bishop set, or extensional set. Setoids are studied
Setoid
but not second countable, hence not metrizable; its metrizable subspaces are all countable. It is hereditarily Lindelöf, hereditarily separable, and perfectly
Split_interval
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
Theorem in set theory
In set theory, Kőnig's theorem states that if the axiom of choice holds, I is a set, κ i {\displaystyle \kappa _{i}} and λ i {\displaystyle \lambda _{i}}
Kőnig's_theorem_(set_theory)
Order whose elements are all comparable
topology induced by a total order may be shown to be hereditarily normal. A totally ordered set is said to be complete if every nonempty subset that has
Total_order
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
Type of topological space
All order topologies on totally ordered sets are hereditarily normal and Hausdorff. Every regular second-countable space is completely normal, and every
Normal_space
Type of binary relation
relation R is called well-founded (or wellfounded or foundational) on a set or, more generally, a class X if every non-empty subset (or subclass) S ⊆
Well-founded_relation
System of mathematical set theory
Tarski–Grothendieck set theory (TG, named after mathematicians Alfred Tarski and Alexander Grothendieck) is an axiomatic set theory. It is a non-conservative
Tarski–Grothendieck set theory
Tarski–Grothendieck_set_theory
Process of repeating items in a self-similar way
scenario that does not use recursion to produce an answer A recursive step — a set of rules that reduces all successive cases toward the base case. For example
Recursion
Target set of a mathematical function
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 the notation
Codomain
Concept in axiomatic set theory
In many popular versions of axiomatic set theory, the axiom schema of specification, also known as the axiom schema of separation (Aussonderungsaxiom)
Axiom_schema_of_specification
Logical connective AND
{\displaystyle \wedge } is the most modern and widely used. The and of a set of operands is true if and only if all of its operands are true, i.e., A
Logical_conjunction
Mathematical concept
elements of some set S {\displaystyle S} have a notion of equivalence (formalized as an equivalence relation), then one may naturally split the set S {\displaystyle
Equivalence_class
3-volume treatise on mathematics, 1910–1913
strings; this form of notation is called an "axiom schema" (i.e., there is a countable number of specific forms the notation could take). This can be read in
Principia_Mathematica
Finite ordered list of elements
n-tuple can be formally defined as the image of a function that has the set of the first n natural numbers as its domain (1, 2, ..., n). Tuples may be
Tuple
Mathematical use of "there exists"
represents the (true) statement There exists some n {\displaystyle n} in the set of natural numbers such that n × n = 25 {\displaystyle n\times n=25} . The
Existential_quantification
Property in general topology
of failures: The theorem can fail without the Hausdorff condition; a countable set with at least two points and with the indiscrete topology is perfect
Finite_intersection_property
HEREDITARILY COUNTABLE-SET
HEREDITARILY COUNTABLE-SET
Girl/Female
English
ancient hereditary title used by Ethiopian queens.
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.
Girl/Female
English American
Modern- ancient hereditary title used by Ethiopian queens.
Boy/Male
Assamese, Celebrity, Gujarati, Hindu, Indian, Jain, Kannada, Malayalam, Marathi, Oriya, Punjabi, Sanskrit, Sikh, Tamil, Telugu, Traditional
Sky; Lord of Day; Uncountable; Space
Boy/Male
Norse
Pointable.
Girl/Female
English
Modern- ancient hereditary title used by Ethiopian queens.
Male
Celtic
, hereditary chief or ruler.
Girl/Female
English
ancient hereditary title used by Ethiopian queens.
Girl/Female
English American
ancient hereditary title used by Ethiopian queens.
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
Boy/Male
Hindu, Indian
Un Countable; Multiple; Countless
Boy/Male
Shakespearean
Measure for Measure' A simple constable.
Male
Celtic
, hereditary prince.
Male
Celtic
, hereditary chief or ruler.
Boy/Male
Shakespearean
Love's Labours Lost' A constable.
Male
Celtic
, hereditary king.
Male
Celtic
, hereditary king.
Boy/Male
Shakespearean
Much Ado About Nothing' A Constable.
Boy/Male
Hindu, Indian
Uncountable
HEREDITARILY COUNTABLE-SET
HEREDITARILY COUNTABLE-SET
Boy/Male
Tamil
Solitary
Boy/Male
American, British, English
From Ann's Estate
Girl/Female
Tamil
Anshumi | அநà¯à®·à¯à®®à¯€
Every part/element of the earth
Girl/Female
Arabic, Assamese, Indian, Kannada, Muslim, Tamil
Dust; A Drink; A Drink of Water
Boy/Male
Hindu, Indian, Telugu
Lord Shiva
Surname or Lastname
North German
North German : from a short form of the personal name Bartholomäus (see Bartholomew).English : habitational name from Meaux (pronounced ‘Myoos’) in Humberside, formerly in East Yorkshire. This was named in Old Norse as ‘sandbank pool’, from melr ‘sandbank’, ‘sandhill’ + sær ‘sea’, ‘lake’, and subsequently assimilated by folk etymology to a French place name.
Girl/Female
American, British, English
God is Gracious; Modern Name Based on Jane or Jean; Based on Janai
Boy/Male
Indian
Attractive
Boy/Male
Tamil
Blue
Male
English
Anglicized form of Hebrew Uwriy, URI means "fiery" or "my flame, my light." In the bible, this is the name of several characters, including a prince of Judah.Â
HEREDITARILY COUNTABLE-SET
HEREDITARILY COUNTABLE-SET
HEREDITARILY COUNTABLE-SET
HEREDITARILY COUNTABLE-SET
HEREDITARILY COUNTABLE-SET
n.
The state or quality of being numerable or countable.
n.
The quality or state of being accountable; accountability.
a.
Such as can be mounted.
a.
Transmitted, or capable of being transmitted, as a constitutional quality or condition from a parent to a child; as, hereditary pride, bravery, disease.
a.
Not cogitable; inconceivable.
n.
A petty constable.
v. t.
To submit; to make accountable.
a.
Descended, or capable of descending, from an ancestor to an heir at law; received or passing by inheritance, or that must pass by inheritance; as, an hereditary estate or crown.
a.
Capable of being numbered.
adv.
By inheritance; in an hereditary manner.
v. t.
Accountable; responsible; sensitive.
a.
See Accountable.
adv.
By inheritance.
n.
To appoint hereditary possessor.
n.
An under constable.
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.
a.
Accountable.
n.
Hereditary character, quality, or disposition.
a.
Hereditary; entailed on a family.