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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)
Branch of mathematics that studies sets
Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any
Set_theory
Branch of music theory
One branch of musical set theory deals with collections (sets and permutations) of pitches and pitch classes (pitch-class set theory), which may be ordered
Set_theory_(music)
M of set theory, it is assumed that M is a set model, i.e. the domain of M is a set in V. If the domain of M is a proper class, then M is a class model
Standard_model_(set_theory)
System of mathematical set theory
larger than sets, such as the class of all sets and the class of all ordinals. Morse–Kelley set theory (MK) allows classes to be defined by formulas whose
Von Neumann–Bernays–Gödel set theory
Von_Neumann–Bernays–Gödel_set_theory
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
Axiomatic constructive set theory is an approach to mathematical constructivism following the program of axiomatic set theory. The same first-order language
Constructive_set_theory
System of mathematical set theory
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 Quine
Morse–Kelley_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
Subfield of mathematical logic
In mathematical logic, descriptive set theory is the study of certain classes of subset of the real line and other Polish spaces satisfying some sort of
Descriptive_set_theory
product Class (set theory) Complement (set theory) Complete Boolean algebra Continuum (set theory) Suslin's problem Continuum hypothesis Countable set Descriptive
List_of_set_theory_topics
Collection of objects studied in music theory
In music theory, as in mathematics (see set) and general parlance, a set (pitch set, pitch-class set, set class, set form, set genus, pitch collection)
Set_(music)
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)
Appendix:Glossary of set theory in Wiktionary, the free dictionary. This is a glossary of terms and definitions related to the topic of set theory. Contents:
Glossary_of_set_theory
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
Generalization of "n-th" to infinite cases
In 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
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)
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
Class that is contained in another class
In set theory and its applications throughout mathematics, a subclass is a class contained in some other class in the same way that a subset is a set contained
Subclass_(set_theory)
of set classes, by Forte number. In music theory, a set class (an abbreviation of pitch-class-set class) is an ascending collection of pitch classes, transposed
List_of_set_classes
Set with exactly one element
x)} Df. That is, 1 is the class of singletons. This is definition 52.01 (p. 363 ibid.) Class (set theory) – Collection of sets in mathematics that can be
Singleton_(mathematics)
Branch of algebraic number theory concerned with abelian extensions
In mathematics, class field theory (CFT) is the fundamental branch of algebraic number theory whose goal is to describe all the abelian Galois extensions
Class_field_theory
Mathematical theory of data types
to set theory as a foundation of mathematics. Examples include Alonzo Church's simple theory of types and Per Martin-Löf's intuitionistic type theory. Many
Type_theory
Set of elements common to all of some sets
In set theory, the intersection of two sets A {\displaystyle A} and B , {\displaystyle B,} denoted by A ∩ B , {\displaystyle A\cap B,} is the set containing
Intersection_(set_theory)
Topics referred to by the same term
Class (philosophy), an analytical concept used differently from such group phenomena as "types" or "kinds" Class (set theory), a collection of sets that
Class
Class of alternative set theories
In mathematical logic, positive set theory is the name for a class of alternative set theories in which the axiom of comprehension holds for at least the
Positive_set_theory
Any collection of sets, or subsets of a set
In set theory and related branches of mathematics, family or collection is used to mean set, indexed set, multiset, tuple, or class. It is usually used
Family_of_sets
Axiomatic set theory proposed by Wilhelm Ackermann
Zermelo–Fraenkel set theory (ZF) in that it allows proper classes, that is, objects that are not sets, including a class of all sets. It replaces several
Ackermann_set_theory
Study of geometric properties of sets through measure theory
geometric measure theory (GMT) is the study of geometric properties of sets (typically in Euclidean space) through measure theory. It allows mathematicians
Geometric_measure_theory
Collection of mathematical objects
of sets. Set theory studies possible axiom systems and their consequences. Since the first half of the 20th century, ZFC (Zermelo–Fraenkel set theory with
Set_(mathematics)
Operation selecting specific components or columns from a set, tuple, or relation
In set theory, a projection is one of two closely related types of functions or operations, namely: A set-theoretic operation typified by the j {\displaystyle
Projection_(set_theory)
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)
System of mathematical set theory
of axiomatic set theory is to distinguish sets from proper classes, if only because mathematics is grounded in sets, with proper classes relegated to
S_(set_theory)
All-encompassing set or class
In set theory, universes are often classes that contain (as elements) all sets for which one hopes to prove a particular theorem. These classes can serve
Universe_(mathematics)
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
Study of computable functions and Turing degrees
computability theory overlaps with proof theory and effective descriptive set theory. Basic questions addressed by computability theory include: What
Computability_theory
Mathematical structure in category theory
category theory, a small set is one in a fixed universe of sets (as the word universe is used in mathematics in general). Thus, the category of small sets is
Small_set_(category_theory)
Classes of partial recursive functions
computability theory, index sets describe classes of computable functions; specifically, they give all indices of functions in a certain class, according
Index_set_(computability)
Theory in human biology
Set point theory, as it pertains to human body weight, states that there is a biological control method in humans that actively regulates weight towards
Set_point_theory
Set of all pitches that are a whole number of octaves apart
octaves. "The pitch class C stands for all possible Cs, in whatever octave position." Important to musical set theory, a pitch class is "all pitches related
Pitch_class
Class of mathematical set whose elements are all subsets
In set theory, a branch of mathematics, a set A {\displaystyle A} is called transitive if either of the following equivalent conditions holds: whenever
Transitive_set
Any one of the distinct objects that make up a set in set theory
"Set Theory", Stanford Encyclopedia of Philosophy, Metaphysics Research Lab, Stanford University Suppes, Patrick (1972) [1960], Axiomatic Set Theory,
Element_of_a_set
Approach to quantum gravity using discrete spacetime
2008 Class. Quantum Grav. 25 202001; arXiv:0806.3083 (Quantum Field Theory) S. Johnston; The Feynman propagator for a Free Scalar Field on a Causal Set; Phys
Causal_sets
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
non-abelian class field theory is a catchphrase, meaning the extension of the results of class field theory, the relatively complete and classical set of results
Non-abelian class field theory
Non-abelian_class_field_theory
In set theory, the critical point of an elementary embedding of a transitive class into another transitive class is the smallest ordinal which is not
Critical_point_(set_theory)
Mathematical concept
equivalence classes to scheme theory Setoid – Mathematical construction of a set with an equivalence relation Transversal (combinatorics) – Set that intersects
Equivalence_class
Concept in axiomatic set theory
axiomatic set theory, the axiom schema of specification, also known as the axiom schema of separation (Aussonderungsaxiom), subset axiom, axiom of class construction
Axiom_schema_of_specification
Area of mathematical logic
theory. The relative emphasis placed on the class of models of a theory as opposed to the class of definable sets within a model fluctuated in the history
Model_theory
Size of a set in mathematics
universe of all sets, the class of all cardinal numbers, and the class of all ordinal numbers are proper classes. Such set theories include Von Neumann–Bernays–Gödel
Cardinality
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 be
Universal_set
Supposition or system of ideas intended to explain something
A theory is, in general, a set of propositions or ideas about something, developed in a variety of ways through any sort of reasoning. This includes informal
Theory
Mathematical logic concept
Shoenfield (1961), establishes the absoluteness of a large class of formulas between a model of set theory and its constructible universe, with important methodological
Absoluteness_(logic)
Generalization of Turing computability
set theory such as Kripke–Platek set theory. It is an important tool in effective descriptive set theory. The central focus of hyperarithmetic theory
Hyperarithmetical_theory
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
1960 mathematics textbook by Paul Halmos
Naive Set Theory is a mathematics textbook by Paul Halmos providing an undergraduate introduction to set theory. Originally published by Van Nostrand
Naive_Set_Theory_(book)
System of mathematical set theory
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 (ZF)
Zermelo_set_theory
concepts in set theory. The implementation of a number of basic mathematical concepts is carried out in parallel in ZFC (the dominant set theory) and in NFU
Implementation of mathematics in set theory
Implementation_of_mathematics_in_set_theory
Concept in set theory
In set theory, the axiom schema of replacement is a schema of axioms in Zermelo–Fraenkel set theory (ZF) that asserts that the image of any set under any
Axiom_schema_of_replacement
Category whose objects are sets and whose morphisms are functions
field of category theory, the category of sets, denoted by Set, is the category whose objects are sets. The arrows or morphisms between sets A and B are the
Category_of_sets
Set of sentences in a formal language
In mathematical logic, a theory (also called a formal theory) is a set of sentences in a formal language. In most scenarios a deductive system is first
Theory_(mathematical_logic)
System of mathematical set theory
In set theory, a semiset is a proper class that is a subclass of a set. In the typical foundations of Zermelo–Fraenkel set theory, semisets are impossible
Semiset
Partition into two separate parts
dichotomy at Wiktionary Binary opposition Bipartite (disambiguation) Class (set theory) Dichotomy paradox Dilemma Dualism Law of excluded middle, which in
Dichotomy
Mathematical set of all subsets of a set
mathematics, the power set (or powerset) of a set S is the set of all subsets of S, including the empty set and S itself. In axiomatic set theory (as developed
Power_set
Study of the practices and possibilities of music
One branch of musical set theory deals with collections (sets and permutations) of pitches and pitch classes (pitch-class set theory), which may be ordered
Music_theory
Mathematician (1845–1918)
mathematician who played a pivotal role in the creation of set theory, which has become a fundamental theory in mathematics. Cantor established the importance
Georg_Cantor
System of mathematical set theory
General set theory (GST) is George Boolos's (1998) name for a fragment of the axiomatic set theory Z. GST is sufficient for all mathematics not requiring
General_set_theory
Psychological concept
neutral set point after a significantly emotional life event. In the literature review, "Beyond the Hedonic Treadmill, Revising the Adaptation Theory of Well-Being"
Hedonic_treadmill
Set whose elements all belong to another set
of k {\displaystyle k} -subsets of an n {\displaystyle n} -element set. In set theory, the notation [ A ] k {\displaystyle [A]^{k}} is also common, especially
Subset
Particular class of sets which can be described entirely in terms of simpler sets
in set theory, the constructible universe (or Gödel's constructible universe), denoted by L , {\displaystyle L,} is a particular class of sets that
Constructible_universe
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
Topics referred to by the same term
(computing), automatic memory management method Set (mathematics) Class (set theory) Family of sets Indexed family Multiset Parametric family Collection
Collection
Equivalence relation expressing that two elements have the same image under a function
In set theory, the kernel of a function f {\displaystyle f} (or equivalence kernel) may be taken to be either the equivalence relation on the function's
Kernel_(set_theory)
Mathematical concept for comparing objects
relation provides a partition of the underlying set into disjoint equivalence classes. Two elements of the given set are equivalent to each other if and only
Equivalence_relation
Notion in computational learning
statistical computational learning theory. Suppose A is a set and C is a class of sets. The class C shatters the set A if for each subset a of A, there
Shattered_set
Paradox in set theory
Russell's paradox. The term "naive set theory" is used in various ways. In one usage, naive set theory is a formal theory, that is formulated in a first-order
Russell's_paradox
Technique invented by Paul Cohen for proving consistency and independence results
In set theory, forcing is a technique for proving consistency and independence results. Intuitively, forcing can be thought of as a technique to expand
Forcing_(mathematics)
Subfield of mathematics
Major subareas include model theory, proof theory, set theory, and recursion theory (also known as computability theory). Research in mathematical logic
Mathematical_logic
Elements in exactly one of two sets
of sets Boolean function Complement (set theory) Difference (set theory) Exclusive or Fuzzy set Intersection (set theory) Jaccard index List of set identities
Symmetric_difference
Mathematical ways to group elements of a set
is sometimes called a setoid, typically in type theory and proof theory. A partition of a set X is a set of non-empty subsets of X such that every element
Partition_of_a_set
Concept in mathematical logic
In set theory, Θ {\displaystyle \Theta } (pronounced like the letter theta) is the least nonzero ordinal α {\displaystyle \alpha } such that there is no
Theta_(set_theory)
Unrelated vertices in graphs
graph theory, an independent set, stable set, coclique or anticlique is a set of vertices in a graph, no two of which are adjacent. That is, it is a set S
Independent set (graph theory)
Independent_set_(graph_theory)
Sequence of words formed by specific rules
computational complexity theory, decision problems are typically defined as formal languages, and complexity classes are defined as the sets of the formal languages
Formal_language
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
3-volume treatise on mathematics, 1910–1913
sections ✱20 GENERAL THEORY OF CLASSES and ✱21 GENERAL THEORY OF RELATIONS. "Relations" are what is known in contemporary set theory as sets of ordered pairs
Principia_Mathematica
Post-tonal music compositional technique
in tone-clock terminology). Tone-clock theory is also concerned with the way that the three-note pitch-class sets (trichords or "triads" in tone-clock terminology)
Tone_clock
System of mathematical set theory
Kripke–Platek set theory (KP), pronounced /ˈkrɪpki ˈplɑːtɛk/, is an axiomatic set theory developed by Saul Kripke and Richard Platek. The theory can be thought
Kripke–Platek_set_theory
product Class (set theory) Complement (set theory) Complete Boolean algebra Continuum (set theory) Suslin's problem Continuum hypothesis Countable set Descriptive
List of mathematical logic topics
List_of_mathematical_logic_topics
In music theory, equivalence class is an equality (=) or equivalence between properties of sets (unordered) or twelve-tone rows (ordered sets). A relation
Equivalence_class_(music)
Kind of proposition in mathematics
In set theory, a branch of mathematics, a reflection principle says that it is possible to find sets that, with respect to any given property, resemble
Reflection_principle
Paradox in set theory
handled in axiomatic set theory by declaring that this collection is not a set but a proper class; in von Neumann–Bernays–Gödel set theory it follows from
Cantor's_paradox
Non-contradiction of a theory
enough fragment of arithmetic—including set theories such as Zermelo–Fraenkel set theory (ZF). These set theories cannot prove their own Gödel sentence—provided
Consistency
Complexity class used to classify decision problems
computational complexity theory, NP (nondeterministic polynomial time) is a complexity class used to classify decision problems. NP is the set of decision problems
NP_(complexity)
Possible axiom of set theory
In set theory, the axiom of limitation of size was proposed by John von Neumann in his 1925 axiom system for sets and classes. It formalizes the limitation
Axiom_of_limitation_of_size
Minimal standard model of ZFC
In set theory, a branch of mathematics, the minimal model is the minimal standard model of ZFC. The minimal model was introduced by Shepherdson (1951,
Minimal_model_(set_theory)
Concept in musical set theory
musical set theory, there are four kinds of interval: Ordered pitch interval Unordered pitch interval Ordered pitch-class interval Unordered pitch-class interval
Pitch_interval
Hierarchy of classes of formal grammars
hierarchy in the fields of formal language theory, computer science, and linguistics, is a containment hierarchy of classes of formal grammars. A formal grammar
Chomsky_hierarchy
Concerned with the notion of stability in model theory
modern model theory and there is a rich framework and set of tools to analyze them. A major direction in model theory is "neostability theory," which tries
Stable_theory
Size of a possibly infinite set
a larger cardinal 2κ). In fact, the class of cardinals is a proper class. (This proof fails in some set theories, notably New Foundations.) All the remaining
Cardinal_number
Set of problems in computational complexity theory
In computational complexity theory, a complexity class is a set of computational problems "of related resource-based complexity". The two most commonly
Complexity_class
CLASS SET-THEORY
CLASS SET-THEORY
Surname or Lastname
English
English : variant spelling of See.
Girl/Female
Indian
Glass
Male
German
Short form of German Niclaus, CLAUS means "victor of the people."Â
Girl/Female
Tamil
Glass
Male
English
Short form of English Stephen, STE means "crown."
Male
Hebrew
Variant spelling of Hebrew Sheth, SHET means "buttocks."
Boy/Male
Australian, Dutch, German, Greek
People's Victory
Surname or Lastname
English and German
English and German : metonymic occupational name for a glazier or glass blower, from Old English glæs ‘glass’ (akin to Glad, referring originally to the bright shine of the material), Middle High German glas.Irish and Scottish : Anglicized form of the epithet glas ‘gray’, ‘green’, ‘blue’ or any of various Gaelic surnames derived from it.German : altered form of the personal name Klass, a reduced form of Nikolaus (see Nicholas).Jewish (Ashkenazic) : ornamental name from German Glass ‘glass’, or a metonymic occupational name for a glazier or glass blower.
Surname or Lastname
English
English : variant of Close 1.German : variant of Kloss.
Surname or Lastname
English
English : from the medieval female personal name Cass, a short form of Cassandra. This was the name (of uncertain, possibly non-Greek, origin) of an ill-fated Trojan prophetess of classical legend, condemned to foretell the future but never be believed; her story was well known and widely popular in medieval England.
Girl/Female
Indian
Glass
Surname or Lastname
North German
North German : topographic name from Middle Low German plas ‘place’, ‘open square’, ‘street’.South German (also Pläss) : from a short form of the medieval personal name Blasius.English : variant of Place 3.
Boy/Male
Australian, Danish, Dutch, Greek, Swedish
People of Victory; Victory of the People
Boy/Male
Greek Latin
People's victory.
Female
Egyptian
, an uncertain goddess.
Male
Hindi/Indian
(सेठ) Hindi name derived from the Sanskrit word setu, SETH means "bridge." Compare with other forms of Seth.
Surname or Lastname
English
English : from the medieval personal name Classe, a short form of Nicholas. See also Clayson.Variant of Klaas or Klass, North German forms of Claus.
Surname or Lastname
English
English : nickname from Old French, Middle English cras ‘big’, ‘fat’ (Latin crassus).Possibly an altered spelling of German Krass.
Boy/Male
Egyptian Hebrew Swedish
Son of Seb and Nut.
Female
English
English short form of Latin Cassandra, CASS means "she who entangles men."Â
CLASS SET-THEORY
CLASS SET-THEORY
Girl/Female
Hindu
Shawl
Surname or Lastname
English
English : variant of Paul.
Boy/Male
Hindu, Indian, Sanskrit
Father; The Middle Child
Boy/Male
Muslim/Islamic
Decorated king
Boy/Male
Indian, Telugu
Lot of Wealth
Boy/Male
Hindu, Indian, Tamil, Traditional
Young Krishna
Girl/Female
Muslim
Diminutive of Nasiba, Noble
Girl/Female
Muslim/Islamic
Beautiful
Boy/Male
Tamil
Expert
Boy/Male
Native American
Rippling brook.
CLASS SET-THEORY
CLASS SET-THEORY
CLASS SET-THEORY
CLASS SET-THEORY
CLASS SET-THEORY
v. t.
An optical glass; a lens; a spyglass; -- in the plural, spectacles; as, a pair of glasses; he wears glasses.
n.
To divide into classes, as students; to form into, or place in, a class or classes.
v. t.
Anything made of glass.
v. t.
To cover or furnish with glass; to glaze.
a.
Regular; uniform; formal; as, a set discourse; a set battle.
n.
A set; a kind or description, species or variety.
a.
Of the rank or degree below the best highest; inferior; second-rate; as, a second-class house; a second-class passage.
n.
One of the sections into which a church or congregation is divided, and which is under the supervision of a class leader.
n.
A group of individuals ranked together as possessing common characteristics; as, the different classes of society; the educated class; the lower classes.
a.
Of the best class; of the highest rank; in the first division; of the best quality; first-rate; as, a first-class telescope.
v. t.
To fix, as a precious stone, in a border of metal; to place in a setting; hence, to place in or amid something which serves as a setting; as, to set glass in a sash.
v. t.
To shut or fasten together with, or as with, a clasp; to shut or fasten (a clasp, or that which fastens with a clasp).
n.
To arrange in classes; to classify or refer to some class; as, to class words or passages.
v. t.
To smooth or polish anything, as leater, by rubbing it with a glass burnisher.
imp. & p. p.
of Set
v. t.
A looking-glass; a mirror.
v. t.
Variant of Clasp
v. t.
To case in glass.