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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)
Substructure of a set theoretical universe
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
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
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
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
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
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
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)
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 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)
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)
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
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
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
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)
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
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)
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
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
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
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)
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
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
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
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)
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)
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
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)
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
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)
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
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
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
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
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
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
Mathematical concept
equivalence classes to scheme theory Setoid – Mathematical construction of a set with an equivalence relation Transversal (combinatorics) – Set that intersects
Equivalence_class
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
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)
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
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
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
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
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)
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
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
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)
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)
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)
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)
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
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
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 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
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
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
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
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
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
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)
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)
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
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
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
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
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
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
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
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)
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
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
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
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
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
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
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
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)
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
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
Alternative to the standard Zermelo–Fraenkel set theory
Internal set theory Pocket set theory Naive set theory S (set theory) Double extension set theory Kripke–Platek set theory Kripke–Platek set theory with urelements
List of alternative set theories
List_of_alternative_set_theories
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
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
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
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
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
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
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
Study of objects of arithmetic interest over infinite towers of number fields
as a Galois module theory of ideal class groups, initiated by Kenkichi Iwasawa (1959) (岩澤 健吉), as part of the theory of cyclotomic fields. In the early
Iwasawa_theory
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
Family of subsets representing "large" sets
topology, including set theory, mathematical logic, model theory (ultraproducts for example), abstract algebra, and others. Filters on a set were later generalized
Filter_on_a_set
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