Zermelo–Fraenkel set theory

Definition

In set theory, Zermelo–Fraenkel set theory, named after mathematicians Ernst Zermelo and Abraham Fraenkel, is an axiomatic system that was proposed in the early twentieth century in order to formulate a theory of sets free of paradoxes such as Russell's paradox. Today, Zermelo–Fraenkel set theory, with the historically controversial axiom of choice (AC) included, is the standard form of axiomatic set theory and as such is the most common foundation of mathematics. Zermelo–Fraenkel set theory with the axiom of choice included is abbreviated ZFC, where C stands for "choice", and ZF refers to the axioms of Zermelo–Fraenkel set theory with the axiom of choice excluded.

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A K Peters Abraham Fraenkel Abstract logic Ackermann set theory Aleph number Alexander Abian Alfred Tarski Algebra of sets Algebraic logic Almost Alphabet (formal languages)Alternative set theory Amorphous set Argument Arity 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 empty set 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 the empty set Axiom of union Axiom schema Axiom schema of replacement Axiom schema of separation Axiom schema of specification Axiomatic set theories Axiomatic set theory Axiomatic system Axiomatization of Boolean algebras Azriel Lévy Banach–Tarski paradox Bertrand Russell Bijection Binary operation Binary relation Boolean algebra Boolean algebras canonically defined Boolean function Burali-Forti paradox Cambridge University Press Cantor's diagonal argument Cantor's paradox Cantor's theorem Cardinal number Cardinality Cartesian product Categorical theory Category (mathematics)Category of sets Category theory Choice function Church encoding Church–Turing thesis Class (set theory)Classical logic Codomain Compactness theorem Complement (set theory)Complete theory Computability theory Computable function Computable set Computably enumerable set Concrete category Congruence modulo n Conservative extension Consistency Consistency proof Constructible universe Construction of the real numbers Constructive mathematics Constructive set theory Continuum hypothesis Countable set De Morgan's laws Decidability (logic)Decision problem Dedekind-infinite set Deductive system Definitional extension Diagram (mathematical logic)Diamondsuit Disjoint sets Disjoint union Domain of a function Domain of discourse Edward N. Zalta Element (mathematics)Elementary diagram Elementary equivalence Elementary function arithmetic Elsevier Empty set Encyclopedia of Mathematics Enumeration Equality (mathematics)Equiconsistency Equivalence relation Eric W. Weisstein Ernst Zermelo Euclid's Elements Euclidean geometry European Mathematical Society 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 model theory Finite set First-order logic First order logic Fixed-point logic Forcing (mathematics)Formal grammar Formal language Formal proof Formal semantics (logic)Formal system Formation rule Foundations of geometry Foundations of mathematics Free logic Free variable Free variables and bound variables Function (mathematics)Functional completeness Functional predicate Fundamenta Mathematicae Fuzzy set Gaisi Takeuti General set theory Generalized Continuum Hypothesis Georg Cantor Grothendieck universe Ground expression Ground formula Gödel's completeness theorem Gödel's incompleteness theorems Gödel's second incompleteness theorem Gödel numbering Halting problem Harvard University Press Heinz-Dieter Ebbinghaus Hereditarily finite set Hereditary set Higher-order logic Hilbert's axioms Hilbert system History of logic History of mathematical logic History of set theory Image (mathematics)Inaccessible cardinal Independence (mathematical logic)Inference Infinite-valued logic Infinite set Information theory Inhabited set Injective function Inner model Internal set theory Interpretation (logic)Interpretation (model theory)Interpretation function Intersection (set theory)Isomorphism Jean van Heijenoort Joan Bagaria John von Neumann Jon Barwise Joseph R. Shoenfield Keith Devlin Kenneth Kunen Kolmogorov complexity Kripke's theory of truth Kripke–Platek set theory Kurepa tree Kurt Gödel Lambda calculus Large cardinal Large cardinal axiom Large cardinals Least element Lemma (mathematics)Lindström's theorem Linear order List of Hilbert systems List of axioms List of first-order theories List of formal systems List of mathematical theories List of set identities and relations List of statements independent of ZFC List of statements undecidable in ZFC Logic Logical biconditional Logical conjunction Logical connective Logical consequence Logical constant Logical disjunction Logical equality Logical equivalence Logical independence Logical truth Logicism Löwenheim–Skolem theorem Many-sorted first-order logic Many-valued logic Map (mathematics)Martin's axiom Material conditional MathWorld Mathematical logic Mathematical object Mathematische Annalen Metalanguage Metamath Metamathematics Metavariable Minimal axioms for Boolean algebra Mizar system Model complete theory Model theory Monadic predicate calculus Monadic second-order logic Moore space (topology)Morse–Kelley set theory Multiverse (set theory)NP (complexity)Naive set theory Natural deduction Natural numbers Negation New Foundations Non-Euclidean geometry Non-logical symbol Non-standard model Non-standard model of arithmetic North-Holland Publishing Company Open formula Operation (mathematics)Ordered pair Ordinal analysis Ordinal number P (complexity)P versus NP problem Paradoxes of set theory Partition of a set Patrick Suppes Paul Bernays Paul Cohen Peano arithmetic Peano axioms Pergamon Press Petr Hájek Philosophy of mathematics Power set Predicate (mathematical logic)Predicate logic Predicate variable Prime model Primitive recursive arithmetic Primitive recursive function Principia Mathematica Projective determinacy Proof of impossibility Proof theory Proper class Proper classes Proposition Propositional calculus Propositional formula Propositional variable Pure set Quanta Magazine Quantifier (logic)Quantifier rank Range of a function Rank (set theory)Recursion Recursive set Relation (mathematics)Reverse mathematics Richard Dedekind Richard Montague Robinson arithmetic 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 of logic Sentence (mathematical logic)Sequent calculus Set-builder notation Set (mathematics)Set builder notation Set membership Set theory Signature (logic)Signature (mathematical logic)Singleton (mathematics)Skolem Skolem arithmetic Solomon Feferman Soundness Spectrum of a sentence Spectrum of a theory Springer-Verlag Springer Science+Business Media Square of opposition Stanford Encyclopedia of Philosophy Strength (mathematical logic)String (formal languages)Strongly inaccessible cardinal Structure (mathematical logic)Subset Substitution (logic)Substructure (mathematics)Subtheory 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 The cumulative hierarchy Theorem Theories of truth Theory (mathematical logic)Theory of sets Thomas Jech Thoralf Skolem Three-valued logic Timeline of mathematical logic Transfer principle Transfinite induction Transitive set True arithmetic Truth predicate Truth table Truth value Tuple 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 Universal quantification Universal set Universe (mathematics)Universe of discourse Unrestricted comprehension Urelement Validity (logic)Variable (mathematics)Venn diagram Von Neumann ordinal Von Neumann universe Von Neumann–Bernays–Gödel set theory Well-formed formula Well-formed formulae Well-founded Well-founded set Well-order Well-ordering theorem Whitehead problem Willard Van Orman Quine Yehoshua Bar-Hillel ZFC (disambiguation)Zermelo Zermelo set theory Zorn's lemma

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