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Hilbert system

Definition

In logic, more specifically proof theory, a Hilbert system, sometimes called Hilbert calculus, Hilbert-style system, Hilbert-style proof system, Hilbert-style deductive system or Hilbert–Ackermann system, is a type of formal proof system attributed to Gottlob Frege and David Hilbert. These deductive systems are most often studied for first-order logic, but are of interest for other logics as well.

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1879 Abstract logic Ackermann set theory Aleph number Alfred Tarski Algebraic logic Alonzo Church Alphabet (formal languages)Ancient Greek Anne Sjerp Troelstra Argument Arity Atomic formula Atomic model (mathematical logic)Atomic sentence Automata theory Automated theorem proving Axiom Axiom of choice Axiom of reducibility Axiom schema Axiomatic system Axiomatization of Boolean algebras Banach–Tarski paradox Begriffsschrift Bijection Binary operation Boolean algebra Boolean algebras canonically defined Boolean function Cantor's diagonal argument Cantor's paradox Cantor's theorem Cardinality Cartesian product Categorical theory Category (mathematics)Category of sets Category theory Church encoding Church–Turing thesis Class (set theory)Classical logic Classical predicate logic Classical propositional 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 David Hilbert Decidability (logic)Decision problem Deduction theorem Deductive system Dependent type Descriptive set theory Determinacy Diagram (mathematical logic)Domain of a function Element (mathematics)Elementary diagram Elementary equivalence Elementary function arithmetic Empty set Enumeration Equiconsistency Equivalence relation Euclid Euclid's Elements Euclidean geometry Existential quantification Expression (mathematics)Extension by definitions Extension by new constant and function names Extensionality 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 variable Free variables and bound variables Frege system Function (mathematics)Functional completeness Functional predicate Fuzzy set General set theory Girard's paradox Glossary of category theory Gottlob Frege Grothendieck universe Ground expression Ground formula Group theory Gödel's completeness theorem Gödel's incompleteness theorems Gödel numbering Halting problem Hereditary set Higher-order logic Higher category theory Hilbert's axioms Hilbert-Lewis system History of logic History of mathematical logic History of type theory Homotopy type theory Image (mathematics)Implicational propositional calculus 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)Intuitionistic logic Intuitionistic predicate logic Intuitionistic type theory Isomorphism Jan Łukasiewicz John von Neumann Judgment (mathematical logic)Kolmogorov complexity Kripke's theory of truth Kripke–Platek set theory Lambda calculus Large cardinal Lemma (mathematics)Lindström's theorem 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 Logic Logical axiom 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)Material conditional Mathematical constructivism Mathematical induction Mathematical logic Mathematical object Mathematical proof Metalanguage Metamath Minimal axioms for Boolean algebra Minimal logic Modal logic Model complete theory Model theory Modus ponens Monadic predicate calculus Monadic second-order logic Morse–Kelley set theory NP (complexity)Naive set theory Natural deduction Necessitation rule Negation New Foundations Non-Euclidean geometry Non-logical symbol Non-standard model Non-standard model of arithmetic Open formula Operation (mathematics)Ordinal analysis Ordinal number P (complexity)P versus NP problem Paradoxes of set theory Partition of a set Peano axioms Philosophy of mathematics Polish notation Power set Predicate (mathematical logic)Predicate logic Predicate variable Prime model Primitive recursive arithmetic Primitive recursive function Principia Mathematica Principles of Mathematical Logic Proof calculus Proof of impossibility Proof theory Proposition Propositional calculus Propositional formula Propositional logic Propositional variable Quantifier (logic)Quantifier rank Recursion Recursive set Relation (mathematics)Reverse mathematics Robinson arithmetic Rule of inference Russell's paradox Satisfiability Saturated model Schröder–Bernstein theorem Second-order arithmetic Second-order logic 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theory)Ultraproduct Uncountable set Undecidable problem Uniform substitution Uninterpreted function Union (set theory)Uniqueness quantification Univalent foundations Universal generalization Universal quantification Universal set Universe (mathematics)Urelement Validity (logic)Variable (mathematics)Venn diagram Von Neumann universe Von Neumann–Bernays–Gödel set theory Well-formed formula Wilhelm Ackermann Zermelo set theory Zermelo–Fraenkel set theory ∞-groupoid ∞-topos

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