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Set (mathematics)

Definition

In mathematics, a set is a collection of different things; the things are called elements or members of the set and are typically mathematical objects: numbers, symbols, points in space, lines, other geometric shapes, variables, functions, or even other sets.

Related concepts

Abraham Fraenkel Abstract logic Ackermann set theory Addition Additive identity Aleph number Algebra of sets Algebraic logic Algebraic structure Almost Alphabet (formal languages)Alternative set theory Amorphous set Argument Arity Associative Atomic formula Atomic model (mathematical logic)Atomic sentence Automata theory Automated theorem proving Axiom Axiom of adjunction Axiom of choice Axiom of constructibility Axiom of countable choice Axiom of dependent choice Axiom of determinacy Axiom of extensionality Axiom of global choice Axiom of infinity Axiom of limitation of size Axiom of pairing Axiom of power set Axiom of projective determinacy Axiom of reducibility Axiom of regularity Axiom of union Axiom schema Axiom schema of replacement Axiom schema of specification Axiomatic set theory Axiomatic system Axiomatization of Boolean algebras Banach–Tarski paradox Bertrand Russell Bijection Binary operation Boolean algebra Boolean algebra (structure)Boolean algebras canonically defined Boolean function Boolean ring Burali-Forti paradox Cambridge University Press Canonical map Cantor's diagonal argument Cantor's paradise Cantor's paradox Cantor's theorem Cardinal number Cardinality Cardinality of the continuum Cartesian product Categorical theory Category (mathematics)Category of sets Category theory Chain (order theory)Church encoding Church–Turing thesis Class (set theory)Classical logic Codomain Combinatorics Commutative Compactness theorem Complement (set theory)Complete lattice Complete theory Computability theory Computable function Computable set Computably enumerable set Concrete category Conservative extension Consistency Constructible universe Construction of the real numbers Constructive set theory Continuum (set theory)Continuum hypothesis Coproduct Countable set Countably infinite set Curly bracket Curve (mathematics)David Hilbert De Morgan's laws Decidability (logic)Decision problem Dedekind-infinite set Deductive system Dependent type Descriptive set theory Determinacy Diagram (mathematical logic)Dimension (mathematics)Disjoint union Domain of a function Domain of discourse Dover Publications Element (mathematics)Elementary diagram Elementary equivalence Elementary function arithmetic Ellipsis Empty set Enumeration Equality (mathematics)Equiconsistency Equivalence relation Ernst Zermelo Euclid Euclid's Elements Euclid's theorem Euclidean geometry Euclidean space Euler diagram Existential quantification Exponentiation Expression (mathematics)Extension by definitions Extension by new constant and function names Extensionality Family of sets Filter on a set Finitary relation Finite-valued logic Finite model theory Finite set First-order logic Fixed-point logic Forcing (mathematics)Formal grammar Formal language Formal proof Formal semantics (logic)Formal system Formation rule Foundational crisis of mathematics Foundations of geometry Foundations of mathematics Free logic Free variables and bound variables Function (mathematics)Functional notation Functional predicate Fuzzy set General set theory Geometric shape Georg Cantor Girard's paradox Glossary of category theory Graph of a function Grothendieck universe Ground expression Ground formula Gödel's completeness theorem Gödel's incompleteness theorems Gödel numbering Halting problem Hamel basis Harvard University Press Hereditarily finite set Hereditary set Higher-order logic Higher category theory Hilbert's axioms Hilbert system History of logic History of mathematical logic History of type theory Homotopy type theory Ideal (ring theory)Identities (mathematics)If and only if Image (mathematics)Inaccessible cardinal Independence (mathematical logic)Indexed family Inference Infinite-valued logic Infinite set Infinity (mathematics)Information theory Inhabited set Injection (mathematics)Injective function Integer Integers Interpretation (logic)Interpretation (model theory)Interpretation function Intersection (set theory)Interval (mathematics)Intuitionistic type theory Irrational number Isomorphism Jean Van Heijenoort John von Neumann Joseph Dauben Journal für die Reine und Angewandte Mathematik Kolmogorov complexity Kripke's theory of truth Kripke–Platek set theory Kurt Gödel Lambda calculus Large cardinal Lemma (mathematics)Lexicographic order Lindström's theorem Line (geometry)Line segment Linearly independent List of Hilbert systems List of axioms List of category theory topics List of first-order theories List of formal systems List of mathematical logic topics List of mathematical theories List of set identities and relations List of set theory topics List of statements independent of ZFC Locus (mathematics)Logic Logical and Logical biconditional Logical conjunction Logical connective Logical consequence Logical constant Logical disjunction Logical equality Logical equivalence Logical formula Logical or Logical truth Logicism Löwenheim–Skolem theorem Many-valued logic Map (mathematics)Martin's axiom Material conditional Mathematical constructivism Mathematical induction Mathematical logic Mathematical object Mathematical proof Mathematical space Mathematics Maximal element Maximal ideal Mereology Metalanguage Minimal axioms for Boolean algebra Modal logic Model complete theory Model theory Monadic predicate calculus Monadic second-order logic Morse–Kelley set theory Multiplication Multiplicative identity NP (complexity)Naive set theory Natural deduction Natural number Natural numbers Negation New Foundations Non-Euclidean geometry Non-logical symbol Non-standard model Non-standard model of arithmetic Number line One-to-one correspondence Open formula Open interval Operation (mathematics)Order of operations Ordered pair Ordinal analysis Ordinal number P (complexity)P versus NP problem Pairwise disjoint Parabola Paradoxes of set theory Partially ordered set Partition of a set Paul Bernays Paul Cohen Paul Halmos Peano axioms Philosophy of mathematics Plane (mathematics)Point (geometry)Potential infinity Power set Powerset Predicate (mathematical logic)Predicate logic Predicate variable Prime model Prime number Primitive recursive arithmetic Primitive recursive function Principia Mathematica Proof of impossibility Proof theory Proper subset Property (mathematics)Proposition Propositional calculus Propositional formula Propositional variable Quantifier (logic)Quantifier rank Rational number Rational numbers Real line Real number Real numbers Real vector space Recursion Recursive set Relation (mathematics)Reverse mathematics Richard Dedekind Ring (mathematics)Robinson arithmetic Rule of inference Russell's paradox Satisfiability Saturated model Schröder–Bernstein theorem Second-order arithmetic Second-order logic Self-verifying theories Semantic theory of truth Semantics of logic Sentence (mathematical logic)Sequence Sequence (mathematics)Sequent calculus Set-builder notation Set (category theory)Set complement Set exponentiation Set intersection Set theory Set union Signature (logic)Singleton (mathematics)Skolem arithmetic Soundness Space (mathematics)Spectrum of a sentence Spectrum of a theory Square function Square of opposition Strength (mathematical logic)String (formal languages)Structuralism (philosophy of mathematics)Structure (mathematical logic)Subset Substitution (logic)Substructure (mathematics)Supertask Surjective function Suslin's problem Syllogism Symbol (formal)Symmetric difference Syntax (logic)T-schema Tarski's axiomatization of the reals Tarski's axioms Tarski's theory of truth Tarski's undefinability theorem Tarski–Grothendieck set theory Tautology (logic)Term (logic)Term logic Theorem Theories of truth Theory (mathematical logic)Thomas Jech Thoralf Skolem Three-valued logic Timeline of mathematical logic Topos Total order Transfer principle Transfinite induction Transitive set True arithmetic Truth predicate Truth table Truth value Tuple Tuples Turing machine Type (model theory)Type theory Ultrafilter (set theory)Ultrafilter on a set Ultraproduct Uncountable set Undecidable problem Uninterpreted function Union (set theory)Uniqueness quantification Univalent foundations Universal quantification Universal set Universe (mathematics)Upper bound Urelement Validity (logic)Variable (mathematics)Vector space Venn diagram Vertical bar Vertical line test Von Neumann universe Von Neumann–Bernays–Gödel set theory Well-formed formula Well-order Willard Van Orman Quine ZFC Zermelo set theory Zermelo–Fraenkel set theory Zorn's lemma ∞-groupoid ∞-topos

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