blanketglossary

Tarski–Grothendieck set theory

Definition

Tarski–Grothendieck set theory is an axiomatic set theory. It is a non-conservative extension of Zermelo–Fraenkel set theory (ZFC) and is distinguished from other axiomatic set theories by the inclusion of Tarski's axiom, which states that for each set there is a "Tarski universe" it belongs to. Tarski's axiom implies the existence of inaccessible cardinals, providing a richer ontology than ZFC. For example, adding this axiom supports category theory.

Related concepts

Abraham Fraenkel Abstract logic Ackermann set theory Aleph number Alexander Grothendieck Alexandre Grothendieck Alfred Tarski Algebraic logic Almost Alphabet (formal languages)Alternative set theory Amorphous set Andreas Blass 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 union Axiom schema Axiom schema of replacement Axiom schema of specification Axiomatic set theory Axiomatic system Axiomatization of Boolean algebras Banach–Tarski paradox Benedikt Löwe Bertrand Russell Bijection Binary operation Boolean algebra Boolean algebras canonically defined Boolean function Burali-Forti paradox Cantor's diagonal argument Cantor's paradox Cantor's theorem Cardinal number Cardinality Cartesian product Categorical theory Category (mathematics)Category of sets Category theory 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 Conservative extension Consistency Constructible universe Construction of the real numbers Constructive set theory Continuum hypothesis Countable set De Morgan's laws Decidability (logic)Decision problem Dedekind-infinite set Deductive system Definition Diagram (mathematical logic)Disjoint union Domain of a function Element (mathematics)Elementary diagram Elementary equivalence Elementary function arithmetic Empty set Enumeration Equiconsistency Equivalence relation Ernst Zermelo Euclid's Elements Euclidean geometry 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 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 variables and bound variables Function (mathematics)Functional predicate Fundamenta Mathematicae Fuzzy set General set theory Georg Cantor Grothendieck universe Ground expression Ground formula Gödel's completeness theorem Gödel's incompleteness theorems Gödel numbering Halting problem Hereditarily finite set Hereditary set Higher-order logic Hilbert's axioms Hilbert system History of logic History of mathematical logic Image (mathematics)Inaccessible cardinal Independence (mathematical logic)Inference Infinite-valued logic Infinite set Information theory Inhabited set Injective function Interpretation (logic)Interpretation (model theory)Interpretation function Intersection (set theory)Isomorphism Jean-Louis Verdier John von Neumann Kolmogorov complexity Kripke's theory of truth Kripke–Platek set theory Kurt Gödel Lambda calculus Large cardinal Lemma (mathematics)Lindström's theorem 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 Logic Logical biconditional Logical conjunction Logical connective Logical consequence Logical constant Logical disjunction Logical equality Logical equivalence Logical truth Logicism Löwenheim–Skolem theorem Many-valued logic Map (mathematics)Martin's axiom Material conditional Mathematical logic Mathematical object Metalanguage Metamath Michael Artin Minimal axioms for Boolean algebra Mizar system Model complete theory Model theory Monadic predicate calculus Monadic second-order logic Morse–Kelley set theory NP (complexity)Naive set theory Natural deduction Negation New Foundations Nicolas Bourbaki Non-Euclidean geometry Non-conservative extension Non-logical symbol Non-standard model Non-standard model of arithmetic Ontology 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 axioms Philosophy of mathematics PlanetMath Power set Powerset Predicate (mathematical logic)Predicate logic Predicate variable Prime model Primitive recursive arithmetic Primitive recursive function Principia Mathematica Proof of impossibility Proof theory Proof verification Proposition Propositional calculus Propositional formula Propositional variable Quantifier (logic)Quantifier rank Range of a function Recursion Recursive set Relation (mathematics)Reverse mathematics Richard Dedekind Robert Solovay 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)Sequent calculus Set-builder notation Set (mathematics)Set theory Signature (logic)Singleton (mathematics)Skolem arithmetic Soundness Spectrum of a sentence Spectrum of a theory Springer Science+Business Media Square of opposition Strength (mathematical logic)String (formal languages)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 Tautology (logic)Term (logic)Term logic Theorem Theories of truth Theory (mathematical logic)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)Unordered pair Urelement Validity (logic)Variable (mathematics)Venn diagram Von Neumann universe Von Neumann–Bernays–Gödel set theory Well-formed formula Willard Van Orman Quine ZFC Zermelo set theory Zermelo–Fraenkel set theory

143 concepts already in your glossary