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Mathematical logic

Definition

Mathematical logic is the study of formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory. Research in mathematical logic commonly addresses the mathematical properties of formal systems of logic such as their expressive or deductive power. However, it can also include uses of logic to characterize correct mathematical reasoning or to establish foundations of mathematics.

Related concepts

(ε, δ)-definition of limit A. M. Turing A. S. Troelstra ACM Computing Classification System A K Peters Abductive reasoning Abraham Fraenkel Abstract algebra Abstract logic Academic Press Ackermann set theory Alan Turing Aleph number Alfred North Whitehead Alfred Tarski Algebra Algebraic geometry Algebraic logic Algebraic number theory Algebraic topology Algorithm Algorithm design Algorithmic efficiency Algorithmic randomness Alice's Adventures in Wonderland Alonzo Church Alpha recursion theory Alphabet (formal languages)American Mathematical Monthly Amsterdam Analysis of algorithms Analytic geometry Analytic number theory Analytic–synthetic distinction Ancient Roman philosophy Antecedent (logic)Antinomy Application security Applied mathematics Areas of mathematics Argument Argumentation theory Aristotelian logic Arithmetic Arithmetic geometry Arithmetical hierarchy Arithmetices principia, nova methodo exposita Arithmetization of analysis Arity Artificial intelligence Atomic formula Atomic model (mathematical logic)Atomic sentence Augmented reality Augustus De Morgan Automata theory Automated planning and scheduling Automated theorem proving Axiom Axiom of choice Axiom of replacement Axiom schema Axiomatic system Axiomatization of Boolean algebras BHK interpretation Banach–Tarski paradox Begriffsschrift Benson Mates Bernard Bolzano Bertrand Russell Bijection Bijection, injection, and surjection Binary operation Boolean algebra Boolean algebra (structure)Boolean algebras canonically defined Boolean function Boston Burali-Forti paradox Busy beaver Calculus Cambridge, Massachusetts Cambridge University Press Cantor's diagonal argument Cantor's first uncountability proof Cantor's paradox Cantor's theorem Cardinal number Cardinal numbers Cardinality Carl Gustav Hempel Cartesian product Categorical logic Categorical theory Category (mathematics)Category of sets Category theory Cauchy Cesare Burali-Forti Chapman & Hall Charles Sanders Peirce Chrysippus Church encoding Church–Turing thesis Class (set theory)Classical logic Claude Shannon Codomain Combinatorics Combinatory logic Communication protocol Commutative algebra Compactness theorem Compiler construction Complement (set theory)Complete theory Completeness theorem Complex analysis Complex geometry Computability theory Computability theory (computer science)Computable Computable function Computable model theory Computable set Computably enumerable set Computational biology Computational chemistry Computational complexity Computational complexity theory Computational engineering Computational geometry Computational intelligence Computational mathematics Computational physics Computational problem Computational social science Computer accessibility Computer algebra Computer animation Computer architecture Computer data storage Computer graphics Computer hardware Computer network Computer science Computer security Computer vision Computing Computing platform Concrete category Concurrency (computer science)Concurrent computing Consequent Conservative extension Consistency Constructible universe Construction of the real numbers Constructive mathematics Constructive set theory Continuous, nowhere differentiable function Continuous function Continuum hypothesis Contradiction Control flow Control theory Converse (logic)Converse nonimplication Countable Countable set Critical thinking Cross-validation (statistics)Cryptography Cumulative hierarchy Curry–Howard correspondence Cut elimination Cyber-physical system Cyberwarfare Cylindric algebra D. C. Heath and Company Daniel H. H. Ingalls Sr.Data mining Database David Hilbert Decidability (logic)Decidable set Decision problem Decision support system Decision theory Dedekind cuts Deductive closure Deductive reasoning Deductive system Definition Dependability Description Descriptive complexity theory Determinacy Diagram (mathematical logic)Dichotomy Differentiable function Differential equation Differential geometry Differential topology Digital art Digital library Digital marketing Diophantine geometry Discrete geometry Discrete mathematics Distributed artificial intelligence Distributed computing Document management system Domain-specific language Domain of a function Domain of discourse Dordrecht E-commerce Edmund Berkeley Educational technology Electronic design automation Electronic publishing Electronic voting Element (mathematics)Elementary algebra Elementary class Elementary diagram Elementary equivalence Elementary function arithmetic Elementary substructure Elliott Mendelson Elliptic geometry Elsevier Embedded system Emil Leon Post Empty set Encyclopedia of Mathematics Engineering mathematics Enterprise information system Enterprise software Entscheidungsproblem Enumeration Equiconsistency Equivalence relation Ernest Addison Moody Ernst Schröder (mathematician)Ernst Zermelo Euclid Euclid's Elements Euclidean geometry European Mathematical Society Exclusive or Existential quantification Expression (mathematics)Extension by definitions Extension by new constant and function names Extensionality Fagin's theorem Fault tolerance Feasible computability Finitary relation Finite-valued logic Finite geometry Finite model theory Finite set First-order logic Fixed-point logic Forcing (mathematics)Form factor (design)Formal grammar Formal language Formal methods Formal proof Formal semantics (logic)Formal semantics (natural language)Formal system Formation rule Foundations of geometry Foundations of mathematics Fourier series Frederic Fitch Free logic Free variables and bound variables Function (mathematics)Function problem Functional analysis Functional completeness Functional predicate Fundamenta Mathematicae Future of mathematics Fuzzy logic Fuzzy set Game theory General set theory General topology Geographic information system Geomathematics Geometric topology Geometry Georg Cantor Georg Kreisel George Boole George Boolos George Peacock (mathematician)Gerhard Gentzen Giuseppe Peano Glossary of mathematical symbols Gottfried Wilhelm Leibniz Gottlob Frege Graph theory Graphics processing unit Great circle Green computing Grothendieck universe Ground expression Ground formula Group theory Grundlagen der Mathematik Gödel's completeness theorem Gödel's incompleteness theorem Gödel's incompleteness theorems Gödel numbering Gödel–Gentzen negative translation Halting problem Hans Reichenbach Hardware acceleration Hardware security Harmonic analysis Harvard University Press Health informatics Heinrich Scholz Helmut Schwichtenberg Hereditary set Hermann Weyl Heyting algebra Higher-order logic Hilary Putnam Hilbert's axioms Hilbert's problems Hilbert's program Hilbert's tenth problem Hilbert-style deduction system Hilbert system History of logic History of mathematical logic History of mathematics Homological algebra Homotopy theory Human-centered computing Human–computer interaction Hyperarithmetical theory Hypercomplex analysis Image (mathematics)Image compression Impredicativity Inaccessible cardinal Independence (mathematical logic)Independence result Index of logic articles Inductive definition Inductive reasoning Industrial process control Inference Infinitary logic Infinitary logics Infinite-valued logic Infinite set Infinitesimal Informal logic Informal mathematics Information retrieval Information security Information system Information theory Inhabited set Injective function Integer Integrated circuit Integrated development environment Interaction design Interpretation (logic)Interpretation (model theory)Interpretation function Interpreter (computing)Intersection (set theory)Intrusion detection system Intuitionism Intuitionistic logic Isomorphism Israel Journal of Mathematics J. L. Austin Jan Lukasiewicz Jean van Heijenoort Johann Heinrich Lambert John von Neumann Jon Barwise Joseph Henry Woodger Joseph R. Shoenfield Journal für die Reine und Angewandte Mathematik Jozef Maria Bochenski Jules Richard (mathematician)Julia Robinson Karl Menger Karl Weierstrass Knot theory Knowledge representation and reasoning Kolmogorov complexity Kripke's theory of truth Kripke model Kripke–Platek set theory Kurt Gödel Lambda calculus Large cardinal Large numbers Lattice (order)Law of the excluded middle Leipzig Lemma (mathematics)Leopold Kronecker Leopold Löwenheim Lewis Carroll Library (computing)Limitation of size Lindström's theorem Linear algebra Linguistic entailment List of Boolean algebra topics List of Hilbert systems List of axioms List of computability and complexity topics List of computer size categories List of fallacies List of films about mathematicians List of first-order theories List of formal systems List of logic symbols List of logicians List of mathematical logic topics List of mathematical theories List of mathematicians List of paradoxes List of rules of inference List of set identities and relations List of set theory topics List of statements independent of ZFC Lists of mathematics topics Logic Logic (disambiguation)Logic gate Logic in China Logic in Greece Logic in India Logic in Islamic philosophy Logic in computer science Logic programming Logic translation Logical NOR Logical biconditional Logical conjunction Logical connective Logical consequence Logical constant Logical disjunction Logical equality Logical equivalence Logical form Logical system Logical truth Logicism Luitzen Egbertus Jan Brouwer Löb's theorem Löwenheim–Skolem theorem Machine learning Many-valued logic Map (mathematics)Martin Davis (mathematician)Material conditional Material nonimplication Mathematical analysis Mathematical and theoretical biology Mathematical chemistry Mathematical economics Mathematical finance Mathematical geology Mathematical induction Mathematical linguistics Mathematical object Mathematical optimization Mathematical physics Mathematical psychology Mathematical sociology Mathematical software Mathematical statistics Mathematicians Mathematics Mathematics and art Mathematics education Mathematische Annalen Measure (mathematics)Menlo Park, California Mereology Metalanguage Metalogic Metamathematics Michael D. Morley Michael Rathjen Middleware Mineola, New York Minimal axioms for Boolean algebra Mobile computing Modal logic Model checking Model complete theory Model of computation Model theory Modeling language Monadic predicate calculus Monadic second-order logic Moritz Pasch Morley's categoricity theorem Morse–Kelley set theory Multi-task learning Multilinear algebra Multimedia database Multiprocessing Multithreading (computer architecture)Munich NIMPLY gate NP (complexity)Naive set theory Name Natural deduction Natural language processing Natural number Natural numbers Necessity and sufficiency Negation Network architecture Network performance Network scheduler Network security Network service Networking hardware New Foundations New York City Nicolas Bourbaki Nikolai Lobachevsky Non-Euclidean geometry Non-classical logic Non-logical symbol Non-standard model Non-standard model of arithmetic Nonclassical logic Number Number-theoretic function Number theory Numerical analysis O-minimal theory On Formally Undecidable Propositions of Principia Mathematica and Related Systems Open formula Open source Operating system Operation (mathematics)Operations research Order theory Ordinal analysis Ordinal number Organon Oskar Morgenstern Outline of computer science Outline of logic Oxford University Press P (complexity)P versus NP problem Paolo Mancosu Paradox Paradoxes of set theory Parallel computing Parallel postulate Partition of a set Pasch's axiom Paul Bernays Paul Cohen Peano arithmetic Peano axioms Peripheral Philosophical logic Philosophy Philosophy of artificial intelligence Philosophy of logic Philosophy of mathematics Philotheus Boehner Photograph manipulation Polish space Power set Powerset Predicate (mathematical logic)Predicate logic Predicate variable Premise Prime model Primitive recursive arithmetic Primitive recursive function Principia Mathematica Printed circuit board Probability Probability theory Proceedings of the London Mathematical Society Processor (computing)Program semantics Program verification Programming language Programming language theory Programming languages Programming paradigm Programming team Programming tool Proof-theoretic ordinal Proof mining Proof of impossibility Proof theory Proposition Propositional calculus Propositional formula Propositional logic Propositional variable Pyotr Novikov Quantifier (logic)Quantifier elimination Quantifier rank Quantifiers (logic)Quantum computing RAND Corporation Randomized algorithm Real-closed field Real-time computing Real analysis Real line Reason Recreational mathematics Recursion Recursion theory Recursive set Recursively enumerable set Reference Reinforcement learning Relation (mathematics)Rendering (computer graphics)Requirements analysis Reverse mathematics Rhetoric Richard's paradox Richard Dedekind Richard Jeffrey Richard Swineshead Robert Lawson Vaught Robert M. Solovay Robinson arithmetic Rudolf Carnap Rule of inference Russell's paradox Rózsa Péter Santa Monica, California Satisfiability Saturated model Saunders Mac Lane Schröder–Bernstein theorem Scope (logic)Second-order arithmetic Second-order logic Security hacker Security service (telecommunication)Self-verifying theories Semantic theory of truth Semantics (computer science)Semantics of logic Sentence (mathematical logic)Sequent calculus Set (mathematics)Set theory Sheffer stroke Signature (logic)Singapore Singleton (mathematics)Skolem's paradox Skolem arithmetic Social computing Social software Software configuration management Software construction Software deployment Software design Software development Software development process Software engineering Software framework Software maintenance Software quality Software repository Solid modeling Solomon Feferman Soundness Spectrum of a sentence Spectrum of a theory Springer Science+Business Media Square of opposition Stanford Encyclopedia of Philosophy Stanislaw Lesniewski Statement (logic)Statistics Stefan Banach Stephen Cole Kleene Stephen Kleene Stewart Shapiro Stochastic computing Stoicism Strength (mathematical logic)String (formal languages)Structure (mathematical logic)Substitution (logic)Substructure (mathematics)Successor function Supertask Supervised learning Surjective function Syllogism Symbol (formal)Syntax Syntax (logic)System on a chip 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 Theoretical computer science Theories of truth Theory (mathematical logic)Theory of computation Thomas Jech Thoralf Skolem Three-valued logic Tibor Radó Timeline of mathematical logic Timeline of mathematics Topology Topos Transactions of the American Mathematical Society Transfer principle Transfinite induction Transfinite number Transitive set True arithmetic Truth Truth function Truth predicate Truth table Truth value Turing degree Turing machine Type (model theory)Type theory Ubiquitous computing Ulrich Kohlenbach Ultrafilter (set theory)Ultraproduct Uncountable set Undecidable problem Uninterpreted function Union (set theory)Uniqueness quantification Universal algebra Universal logic Universal quantification Universal set Universe (mathematics)Unsupervised learning Urelement Urelements Validity (logic)Variable (mathematics)Vatroslav Bertić Vaught conjecture Venn diagram Very-large-scale integration Victor J. Katz Video game Vienna Virtual machine Virtual reality Visualization (graphics)Von Neumann universe Von Neumann–Bernays–Gödel set theory W. Hugh Woodin Well-formed formula Well-ordered Well-ordering theorem Wilfrid Hodges Wireless sensor network Wolfgang Rautenberg Word problem for groups Word processor World Scientific Publishing World Wide Web XNOR gate Yuri Matiyasevich Zermelo set theory Zermelo–Fraenkel set theory Éléments de mathématique

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