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Formal system

Definition

A formal system is an abstract structure and formalization of an axiomatic system used for deducing, using rules of inference, theorems from axioms.

Related concepts

Abstract logic Abstract structure Ackermann set theory Aleph number Alexander Bogdanov Algebraic logic Allenna Leonard Alphabet (formal languages)Analytic grammar Anatol Rapoport Anthony Wilden Argument Arity Atomic formula Atomic model (mathematical logic)Atomic sentence Augmentation Research Center Augustus De Morgan Automata theory Automated theorem proving Axiom Axiom of choice Axiom of reducibility Axiom schema Axiomatic system Axiomatization of Boolean algebras Axioms Banach–Tarski paradox Barbara J. Grosz Bas van Fraassen Bijection Binary operation Biological system Boolean algebra Boolean algebras canonically defined Boolean function Bra–ket notation Béla H. Bánáthy C. West Churchman Cantor's diagonal argument Cantor's paradox Cantor's theorem Cardinality Cartesian product Categorical theory Category (mathematics)Category of sets Category theory Charles A. S. Hall Church encoding Church–Turing thesis Class (set theory)Classical logic Claude Shannon Codomain Compactness theorem Complement (set theory)Complete theory Completeness (logic)Complex system Computability theory Computable function Computable set Computably enumerable set Computational linguistics Concrete category Conservative extension Consistency Constructible universe Construction of the real numbers Constructive set theory Continuum hypothesis Control theory Countable set Coupled human–environment system Cybernetics David Hilbert Decidability (logic)Decidable set Decision problem Deductive reasoning Deductive system Dependent type Descriptive set theory Determinacy Diagram (mathematical logic)Domain of a function Domain of discourse Donella Meadows Doubling time Douglas Hofstadter Earth system science Economic system Ecosystem Edsger W. Dijkstra Edward Norton Lorenz Element (mathematics)Elementary diagram Elementary equivalence Elementary function arithmetic Empty set Enumeration Equiconsistency Equivalence relation Eric Trist Euclid's Elements Euclidean geometry Europe Existential quantification Expression (mathematics)Extension by definitions Extension by new constant and function names Extensionality Faina Mihajlovna Kirillova Finitary relation Finite-valued logic Finite model theory Finite set First-order logic Fixed-point logic Forcing (mathematics)Formal grammar Formal language Formal languages Formal logical systems Formal method Formal methods Formal proof Formal science Formal semantics (logic)Formal verification Formalism (philosophy of mathematics)Formation rule Foundational crisis of mathematics Foundations of geometry Foundations of mathematics Francisco Varela Fred Emery Free logic Free variables and bound variables Function (mathematics)Functional predicate Fuzzy set General set theory Generative grammar Geoffrey Hunter (logician)George Boole George Dantzig George Klir Girard's paradox Glossary of category theory Gongsun Long Gottlob Frege Gregory Bateson Grothendieck universe Ground expression Ground formula Gödel's completeness theorem Gödel's incompleteness theorems Gödel numbering Halting problem Heinz von Foerster Hereditary set Higher-order logic Higher category theory Hilbert's axioms Hilbert's program Hilbert system History of logic History of mathematical logic History of type theory Homotopy type theory Howard T. Odum Humberto Maturana Ilya Prigogine Image (mathematics)Inaccessible cardinal Independence (mathematical logic)Inference Infinite-valued logic Infinite set Information system Information theory Inhabited set Injective function Intended interpretation Interpretation (logic)Interpretation (model theory)Interpretation function Intersection (set theory)Intuitionistic type theory Isomorphism James Grier Miller James J. Kay Jay Wright Forrester Jeff Rulifson Jennifer Wilby John Seddon Kathleen Carley Katia Sycara Kenneth E. Boulding Kevin Warwick Kolmogorov complexity Kripke's theory of truth Kripke–Platek set theory Kurt Gödel Lambda calculus Large cardinal Lemma (mathematics)Limiting factor Lindström's theorem Linguistics 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 List of systems sciences organizations List of systems scientists Living systems Logic Logic translation Logical biconditional Logical conjunction Logical connective Logical consequence Logical constant Logical disjunction Logical equality Logical equivalence Logical form Logical truth Logicism Ludwig von Bertalanffy Lydia Kavraki Löwenheim–Skolem theorem Manfred Clynes Manuela M. Veloso Many-valued logic Map (mathematics)Margaret Boden Margaret Mead Mary Cartwright Material conditional Mathematical constructivism Mathematical induction Mathematical linguistics Mathematical logic Mathematical notation Mathematical object Mathematical proof Mathematics Metalanguage Metamathematics Metatheorem Mihajlo D. Mesarovic Mike Jackson (systems scientist)Minimal axioms for Boolean algebra Modal logic Model (model theory)Model complete theory Model theory Monadic predicate calculus Monadic second-order logic Morse–Kelley set theory Multi-agent system Murray Bowen NP (complexity)Naive set theory Natural deduction Natural language processing Negation Negative feedback Nervous system New Foundations Niklas Luhmann Non-Euclidean geometry Non-logical axioms Non-logical symbol Non-standard model Non-standard model of arithmetic Nonnegative integer Nonsense Norbert Wiener Notation Open formula Operation (mathematics)Ordinal analysis Ordinal number P (complexity)P versus NP problem Paradoxes of set theory Partition of a set Paul Dirac Peano arithmetic Peano axioms Peter Senge Philosophy of mathematics PlanetMath Positive feedback Power set Predicate (mathematical logic)Predicate logic Predicate variable Prime model Primitive recursive arithmetic Primitive recursive function Principia Cybernetica Principia Mathematica Production (computer science)Programming language theory Proof calculus Proof of impossibility Proof system Proof theory Proposition Propositional calculus Propositional formula Propositional variable Pāṇini QED manifesto Qian Xuesen Quantifier (logic)Quantifier rank Radhika Nagpal Raymond M. Smullyan Recommender system Recursion Recursive set Recursively enumerable set Regular expression Relation (mathematics)Reverse mathematics Rewriting system Richard E. Bellman Robinson arithmetic Rule of inference Rules of inference Russell's paradox Russell L. Ackoff Ruzena Bajcsy Satisfiability Saturated model 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 Set (mathematics)Set theory Signature (logic)Singleton (mathematics)Skolem arithmetic Social system Sociotechnical system Soundness Spectrum of a sentence Spectrum of a theory Square of opposition Stafford Beer Stephanie Forrest Stephen Cole Kleene Strength (mathematical logic)String (computer science)String (formal languages)Structuralism (philosophy of mathematics)Structure (mathematical logic)Substitution (logic)Substitution instance Substructure (mathematics)Supertask Surjective function Syllogism Symbol (formal)Syntax (logic)Syntax analysis System System dynamics Systemics Systems analysis Systems art Systems biology Systems ecology Systems engineering Systems neuroscience Systems pharmacology Systems philosophy Systems psychology Systems science Systems theory Systems theory in anthropology Systems theory in archaeology Systems theory in political science Systems thinking T-schema Talcott Parsons 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 Theory (mathematical logic)Three-valued logic Timeline of mathematical logic Topos Transfer principle Transitive set True arithmetic Truth predicate Truth table Truth value Turing machine Twelve leverage points Type (model theory)Type theory Ultrafilter (set theory)Ultraproduct Uncountable set Undecidable problem Uninterpreted function Union (set theory)Uniqueness quantification Univalent foundations Universal quantification Universal set Universe (mathematics)Urban metabolism Urelement Validity (logic)Variable (mathematics)Venn diagram Victor Aladjev Von Neumann universe Von Neumann–Bernays–Gödel set theory Well-formed formula William Ross Ashby World-systems theory Zermelo set theory Zermelo–Fraenkel set theory ∞-groupoid ∞-topos

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