Set theory

Definition

Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory – as a branch of mathematics – is mostly concerned with those that are relevant to mathematics as a whole.

Related concepts

Abraham Fraenkel Abstract algebra Abstract logic Ackermann set theory Actual infinity Akihiro Kanamori Aleph Aleph number Algebra Algebra of sets Algebraic geometry Algebraic logic Algebraic number theory Algebraic topology Almost Alphabet (formal languages)Alternative set theory Amorphous set Analytic geometry Analytic number theory Andrey Kolmogorov Antinomy Applied mathematics Areas of mathematics Argument Aristotle Arithmetic Arithmetic geometry Arity Arthur Schoenflies 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 regularity Axiom of union Axiom schema Axiom schema of replacement Axiom schema of separation Axiom schema of specification Axiom schema of unrestricted comprehension Axiomatic system Axiomatization of Boolean algebras Banach–Tarski paradox Basic Law V Bernard Bolzano Bernhard Riemann Bertrand Russell Bijection Binary operation Binary relation Boolean-valued model Boolean algebra Boolean algebras canonically defined Boolean function Boolean logic Borel equivalence relation Borel hierarchy Borel set Burali-Forti paradox Calculus Cantor's diagonal argument Cantor's first set theory article Cantor's paradox Cantor's theorem Cardinal arithmetic Cardinal characteristics of the continuum Cardinal number Cardinality Carl Friedrich Gauss Cartesian product Categorical theory Categories (Aristotle)Category (mathematics)Category of sets Category theory Church encoding Church–Turing thesis Circular definition Clarendon Press Class (set theory)Classical logic Codomain Combinatorics Communications on Pure and Applied Mathematics Commutative algebra Compactness theorem Complement (set theory)Complete theory Complex analysis Complex geometry Computability theory Computable function Computable set Computably enumerable set Computational complexity theory Computational geometry Computational mathematics Computer algebra Computer programming Computer science Concrete category Conservative extension Consistency Consistency (mathematical logic)Constructible universe Construction of the real numbers Constructive set theory Constructivism (math)Continuum hypothesis Contradiction Control theory Controversy over Cantor's theory Countable set Crispin Wright Cumulative hierarchy De Morgan's laws Decidability (logic)Decision problem Decision theory Dedekind-infinite set Dedekind cuts Deductive system Descriptive set theory Determinacy Diagram (mathematical logic)Differential equation Differential geometry Differential topology Diophantine geometry Discontinuous function Discrete geometry Discrete mathematics Disjoint union Domain of a function Dover Publications Edward N. Zalta Edward Nelson Effective descriptive set theory Element (mathematics)Elementary Theory of the Category of Sets Elementary algebra Elementary diagram Elementary equivalence Elementary function arithmetic Empty set Encyclopedia of Mathematics Engineering mathematics Enumeration Epsilon Equiconsistency Equinumerous Equivalence relation Equivalence relations Erdős–Rado theorem Ernst Zermelo Errett Bishop Euclid's Elements Euclidean geometry European Mathematical Society Evolutionary dynamics Existential quantification 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 geometry Finite model theory Finite set Finitism First-order logic Fixed-point logic Forcing (mathematics)Formal grammar Formal language Formal proof Formal semantics (logic)Formal system Formation rule Foundation of mathematics Foundational crisis of mathematics Foundations of Arithmetic Foundations of geometry Foundations of mathematics Free logic Free variables and bound variables Function (mathematics)Functional analysis Functional predicate Future of mathematics Fuzzy set Fuzzy set theory Galileo's paradox Game theory General set theory General topology Geomathematics Geometric topology Geometry Georg Cantor Georg Kreisel Giuseppe Peano Glossary of mathematical symbols Glossary of set theory Google Books Gottlob Frege Graph (discrete mathematics)Graph theory Grothendieck universe Ground expression Ground formula Group theory Gödel's completeness theorem Gödel's incompleteness theorems Gödel numbering Halting problem Harmonic analysis Henri Poincaré Hereditarily finite set Hereditary set Hermann Weyl Higher-order logic Higher inductive type Hilbert's axioms Hilbert system History of calculus History of logic History of mathematical logic History of mathematics Homological algebra Homomorphisms Homotopy theory Homotopy type theory Hume's principle Hyperarithmetical theory Hypercomplex analysis Image (mathematics)Impredicativity Inaccessible cardinal Independence (mathematical logic)Indian mathematics Inference Infinitary combinatorics Infinite-valued logic Infinite set Infinity Infinity (philosophy)Informal mathematics Information theory Inhabited set Injective function Inner model theory Institute for Advanced Study Integer Intensional definition Internal set theory Internet Encyclopedia of Philosophy Interpretation (logic)Interpretation (model theory)Interpretation function Intersection (set theory)Interval (mathematics)Intuitionistic logic Invariant (mathematics)Irving Kaplansky Isomorphism Iterative conception of set John Venn John von Neumann Joseph Dauben Journal für die reine und angewandte Mathematik Julius König Keith Devlin Kenneth Kunen Klein's encyclopedia Knot theory Kolmogorov complexity Kripke's theory of truth Kripke–Platek set theory Kurt Gödel L. E. J. Brouwer Lambda calculus Large cardinal Large cardinal property Large numbers Law of the excluded middle Lemma (mathematics)Leopold Kronecker Lightface pointclass Lindström's theorem Linear algebra List (abstract data type)List of Hilbert systems List of axioms List of films about mathematicians List of first-order theories List of formal systems List of mathematical theories List of mathematicians List of set identities and relations List of set theory topics List of statements independent of ZFC Lists of mathematics topics Logic Logical biconditional Logical conjunction Logical connective Logical consequence Logical constant Logical disjunction Logical equality Logical equivalence Logical operators Logical truth Logicism Lotfi A. Zadeh Ludwig Wittgenstein Löwenheim–Skolem theorem Manifold Manifolds Many-valued logic Map (mathematics)Martin's axiom Mary Tiles Material conditional Mathematical analysis Mathematical and theoretical biology Mathematical chemistry Mathematical constructivism Mathematical economics Mathematical finance Mathematical function Mathematical geology Mathematical linguistics Mathematical logic Mathematical object Mathematical optimization Mathematical physics Mathematical platonism Mathematical psychology Mathematical sociology Mathematical statistics Mathematicians Mathematics Mathematics and art Mathematics education Matthew Foreman Meagre set Measurable cardinal Measure (mathematics)Metalanguage Metamath Michael Dummett Minimal axioms for Boolean algebra Model complete theory Model theory Monadic predicate calculus Monadic second-order logic Moore space (topology)Morse–Kelley set theory Multilinear algebra Multiset NLab NP (complexity)Naive Set Theory (book)Naive set theory Natural deduction Natural number Natural numbers Negation New Foundations New Math Non-Euclidean geometry Non-logical symbol Non-standard model Non-standard model of arithmetic Number Number theory Numerical analysis Omega On a Property of the Collection of All Real Algebraic Numbers One-to-one correspondence Ontology Open formula Operation (mathematics)Operations research Order relation Order theory Ordered pair Ordinal analysis Ordinal number Oxford University Press P (complexity)P versus NP problem Paradox Paradoxes Paradoxes of set theory Paradoxes of the Infinite Parallel postulate Partition of a set Paul Bernays Paul Cohen Paul Cohen (mathematician)Paul Halmos Peano arithmetic Peano axioms Philosophical logic Philosophical objections to Cantor's theory Philosophy Philosophy of mathematics Point-set topology Point set Pointclass Pointless topology Polish space Porphyrian tree Power set Predicate (mathematical logic)Predicate logic Predicate variable Primary school Prime model Primitive recursive arithmetic Primitive recursive function Principia Mathematica Probability Probability theory Programming language Projective hierarchy Proof of impossibility Proof theory Proper class Proper subset Property (philosophy)Proposition Propositional calculus Propositional formula Propositional logic Propositional variable Pure set Quantifier (logic)Quantifier rank R. L. Goodstein Ramsey's theorem Range of a function Raymond Smullyan Real analysis Real line Real number Recreational mathematics Recursion Recursion theory Recursive set Relation (mathematics)Relational algebra Relational model Remarks on the Foundations of Mathematics Reverse mathematics Richard Dedekind Riemann integral Ring (mathematics)Robinson arithmetic Rough set Rule of inference Russell's paradox Satisfiability Saturated model Saunders Mac Lane Schröder–Bernstein theorem Second-order arithmetic Second-order logic Self-verifying theories Semantic theory of truth Semantics (computer science)Semantics of logic Sentence (mathematical logic)Sequent calculus Sergei Fomin Set-builder notation Set-theoretic topology Set (abstract data type)Set (mathematics)Set difference Set membership Set theory (disambiguation)Set theory (music)Signature (logic)Singleton (mathematics)Skolem arithmetic Solomon Feferman Soundness Spectrum of a sentence Spectrum of a theory Springer-Verlag Springer Verlag Square of opposition Stanford Encyclopedia of Philosophy Statistics Stone space Strength (mathematical logic)String (formal languages)Structural set theory Structure (mathematical logic)Subset Substitution (logic)Substructure (mathematics)Supertask Surjective function Suslin's problem Syllogism Symbol (formal)Symmetric difference Syntax (logic)Systems science 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)Theory of computation Thomas Forster (mathematician)Thomas Jech Thoralf Skolem Three-valued logic Timeline of mathematical logic Timeline of mathematics Topology Topos theory Transfer principle Transfinite induction Transfinite numbers Transfinite recursion Transitive set Tree of Porphyry Trigonometric series True arithmetic Truth predicate Truth table Truth value Tuple Turing Machine 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 algebra Universal properties Universal quantification Universal set Universe (mathematics)University of Wisconsin-Milwaukee Urelement Validity (logic)Variable (mathematics)Vector space Venn diagram Von Neumann universe Von Neumann–Bernays–Gödel set theory Wadge degree Wadge hierarchy Walter Rudin Well-formed formula Willard Van Orman Quine YouTube ZFC Zeno of Elea Zermelo set theory Zermelo–Fraenkel set theory ℵ

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