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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
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
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)
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)
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)
Subfield of mathematical logic
In mathematics, descriptive set theory is the study of certain classes of subsets of the real line and other Polish spaces satisfying some sort of definability
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
Perspective of mathematical philosophy
In mathematical set theory, the multiverse view is that there are many models of set theory, but no "absolute", "canonical" or "true" model. The various
Multiverse_(set_theory)
Branch of music theory
Musical set theory provides concepts for categorizing musical objects and describing their relationships. Howard Hanson first elaborated many of the concepts
Set_theory_(music)
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)
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
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
Topics referred to by the same term
Look up set theory in Wiktionary, the free dictionary. Set theory is a branch of mathematics concerning mathematical sets. A set theory may also refer
Set_theory_(disambiguation)
The real numbers or their cardinality
In the mathematical field of set theory, the continuum means the real numbers, or the corresponding (infinite) cardinal number, denoted by c {\displaystyle
Continuum_(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
Topics referred to by the same term
wedge product) Internal product, in a monoidal category Product (category theory), a generalization of mathematical products Fibre product or pullback Coproduct
Product
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
System of mathematical set theory
Internal set theory (IST) is a mathematical theory of sets developed by Edward Nelson that provides an axiomatic basis for a portion of the nonstandard
Internal_set_theory
System of mathematical set theory
Neumann–Bernays–Gödel set theory (NBG) is an axiomatic set theory that is a conservative extension of Zermelo–Fraenkel–choice set theory (ZFC). NBG introduces
Von Neumann–Bernays–Gödel set theory
Von_Neumann–Bernays–Gödel_set_theory
Partial order with well-ordered predecessors
In set theory, a tree is a partially ordered set ( T , < ) {\displaystyle (T,<)} such that for each t ∈ T {\displaystyle t\in T} , the set { s ∈ T : s
Tree_(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)
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
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
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
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
In set theory, a mouse is a small model of (a fragment of) Zermelo–Fraenkel set theory with desirable properties. The exact definition depends on the
Mouse_(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)
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
Identities and relationships involving sets
mathematics, particularly in the study of set theory, the algebra of sets defines the properties and laws of sets, the set-theoretic operations of union, intersection
Algebra_of_sets
Axiomatic set theory proposed by Wilhelm Ackermann
mathematics and logic, Ackermann set theory (AST, also known as A ∗ / V {\displaystyle A^{*}/V} ) is an axiomatic set theory proposed by Wilhelm Ackermann
Ackermann_set_theory
Branch of topology
conditions for a topological space to be metrizable. Set-theoretic topology is a subject that combines set theory and general topology. It focuses on topological
General_topology
Technique for proving 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)
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
Area of mathematical logic
the sets that can be defined in a model of a theory, and the relationship of such definable sets to each other. As a separate discipline, model theory goes
Model_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)
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
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
Approach to quantum gravity using discrete spacetime
provides a theory in which space time is fundamentally discrete while retaining local Lorentz invariance. A causal set (or causet) is a set C {\displaystyle
Causal_sets
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
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
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
Branch of theoretical mathematics
Named set theory is a branch of theoretical mathematics that studies the structures of names. The named set is a theoretical concept that generalizes
Named_set_theory
Mathematical set formed from two given sets
In mathematics, specifically set theory, the Cartesian product of two sets A and B, denoted A × B, is the set of all ordered pairs (a, b) where a is an
Cartesian_product
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
3-volume treatise on mathematics, 1910–1913
and set theory at the turn of the 20th century, like Russell's paradox. This third aim motivated the adoption of the theory of types in PM. The theory of
Principia_Mathematica
Size of a set in mathematics
unprovable and undisprovable in standard set theories such as Zermelo–Fraenkel set theory. Alternative set theories and additional axioms give rise to different
Cardinality
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 can
Constructible_universe
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
Collection of prefixes of finite sequences
In descriptive set theory, a tree on a set X {\displaystyle X} is a collection of finite sequences of elements of X {\displaystyle X} such that every
Tree_(descriptive_set_theory)
Axiom of set theory
an axiom of set theory. Informally put, the axiom of choice says that given any collection of non-empty sets, one can identify another set containing one
Axiom_of_choice
Study of discrete mathematical structures
development of the theory of infinite sets is outside the scope of discrete mathematics. Indeed, contemporary work in descriptive set theory makes extensive
Discrete_mathematics
Mathematical models of strategic interactions
game theory include algorithmic game theory, behavioral game theory, combinatorial game theory, evolutionary game theory, and quantum game theory. In 1994
Game_theory
Collection of mathematical objects
ultimately defined in terms of sets. That is, set theory serves as foundation of mathematics. Before the end of the 19th century, sets were not studied specifically
Set_(mathematics)
discrete and Euclidean geometries, graph theory, group theory, mathematical logic, number theory, set theory, Ramsey theory, dynamical systems, and partial differential
List of unsolved problems in mathematics
List_of_unsolved_problems_in_mathematics
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
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
Hypothetical group of multiple universes
relies on string theory's Calabi–Yau spaces. Quantum fluctuations drop the shapes to a lower energy level, creating a pocket with a set of laws different
Multiverse
Mathematical set with an ordering
In mathematics, especially order theory, a partial order on a set is an arrangement such that, for certain pairs of elements, one precedes the other. The
Partially_ordered_set
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
Substructure of a set theoretical universe
In set theory, a standard model for a theory T (in the language of set theory) is a model M for T where the membership relation ∈M is the same as the membership
Standard_model_(set_theory)
Mathematical logic concept
particularly important in set theory and model theory, fields where multiple structures are considered simultaneously. In model theory, several basic results
Absoluteness_(logic)
Sets whose elements have degrees of membership
does not belong to the set. By contrast, fuzzy set theory permits the gradual assessment of the membership of elements in a set; this is described with
Fuzzy_set
Mathematical theory
Quasi-set theory is a formal mathematical theory for dealing with collections of objects, some of which may be indistinguishable from one another. Quasi-set
Quasi-set_theory
Axiom used in set theory
in many forms of axiomatic set theory, such as the Zermelo–Fraenkel set theory. Informally, the axiom means that the two sets A and B are equal if and only
Axiom_of_extensionality
Basic notion of sameness in mathematics
century, set theory (specifically Zermelo–Fraenkel set theory) became the most common foundation of mathematics. In set theory, any two sets are defined
Equality_(mathematics)
Basic framework of mathematics
mathematical logic that includes set theory, model theory, proof theory, computability and computational complexity theory, and more recently, parts of computer
Foundations_of_mathematics
Statement that is taken to be true
Zermelo–Fraenkel set theory with choice, abbreviated ZFC, or some very similar system of axiomatic set theory like Von Neumann–Bernays–Gödel set theory, a conservative
Axiom
Concept in philosophy and set theory
publish a series of papers developing his theory of cardinality (Mächtigkeit, magnitude) of infinite sets. In his first paper, On a Property of the Collection
Absolute_infinite
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
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
descriptive set theory, a scale is a certain kind of object defined on a set of points in some Polish space (for example, a scale might be defined on a set of
Scale (descriptive set theory)
Scale_(descriptive_set_theory)
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
All-encompassing set or class
In mathematics, and particularly in set theory, category theory, type theory, and the foundations of mathematics, a universe is a collection that contains
Universe_(mathematics)
Type of infinite number in set theory
In 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_cardinal
Pair of logical equivalences
output, as well as change the operator when doing a substitution. In set theory, it is often stated as "union and intersection interchange under complementation"
De_Morgan's_laws
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
Ordered listing of items in collection
have little to do with the underlying set, it is useful when some order of the set is necessary. In set theory, there is a more general notion of an enumeration
Enumeration
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
Mathematics textbook
Set Theory: An Introduction to Independence Proofs is a textbook and reference work in set theory by Kenneth Kunen. It starts from basic notions, including
Set Theory: An Introduction to Independence Proofs
Set_Theory:_An_Introduction_to_Independence_Proofs
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
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
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
Overview of and topical guide to logic
Codomain Complement (set theory) Constructible universe Continuum hypothesis Countable set Decidable set Denumerable set Disjoint sets Disjoint union Domain
Outline_of_logic
In set theory, a branch of mathematics, the axiom of uniformization is a weak form of the axiom of choice. It states that if R {\displaystyle R} is a subset
Uniformization_(set_theory)
Algebraic concept in measure theory, also referred to as an algebra of sets
algebras over fields or rings in ring theory. Fields of sets play an essential role in the representation theory of Boolean algebras. Every Boolean algebra
Field_of_sets
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
Set with exactly one element
0} . Within the framework of Zermelo–Fraenkel set theory, the axiom of regularity guarantees that no set is an element of itself. This implies that a singleton
Singleton_(mathematics)
Infinite cardinal number
particularly in set theory, the aleph numbers are a sequence of numbers used to represent the cardinality (or size) of infinite sets. They were introduced
Aleph_number
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
Branch of biology
life processes was developed since 1970 in connection with molecular set theory, relational biology and algebraic biology.[citation needed] A monograph
Mathematical and theoretical biology
Mathematical_and_theoretical_biology
Algebraic manipulation of "true" and "false"
provided for in all modern programming languages. It is also used in set theory and statistics. A precursor of Boolean algebra was Gottfried Wilhelm Leibniz's
Boolean_algebra
Hungarian and American mathematician and physicist (1903–1957)
not a set. Overall, von Neumann's major achievement in set theory was an "axiomatization of set theory and (connected with that) elegant theory of the
John_von_Neumann
Branch of mathematics
upon the concepts of set theory, arithmetic, and binary relations. Orders are special binary relations. Suppose that P is a set and that ≤ is a relation
Order_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)
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)
Symbol connecting formulas in logic
2024-06-11. "1.5 Logic and Sets". www.whitman.edu. Retrieved 2024-06-11. "Theory Set". mirror.clarkson.edu. Retrieved 2024-06-11. "Set Inclusion and Relations"
Logical_connective
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)
System of mathematical set theory
set theory (KP) is an axiomatic set theory developed by Saul Kripke and Richard Platek. It is substantially weaker than Zermelo–Fraenkel set theory (ZF);
Kripke–Platek_set_theory
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
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