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Proof theory

Definition

Proof theory is a major branch of mathematical logic and theoretical computer science within which proofs are treated as formal mathematical objects, facilitating their analysis by mathematical techniques. Proofs are typically presented as inductively defined data structures such as lists, boxed lists, or trees, which are constructed according to the axioms and rules of inference of a given logical system. Consequently, proof theory is syntactic in nature, in contrast to model theory, which is semantic in nature.

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A. S. Troelstra Abstract logic Ackermann set theory Alan Turing Aleph number Algebraic logic Almqvist & Wiksell Alphabet (formal languages)Analytic proof Argument Arithmetical hierarchy Arity Atomic formula Atomic model (mathematical logic)Atomic sentence Automata theory Automated theorem proving Axiom Axiom of choice Axiom schema Axiomatic set theory Axiomatic system Axiomatization of Boolean algebras Banach–Tarski paradox Bertrand Russell Bijection Binary operation Boolean algebra Boolean algebras canonically defined Boolean function Cantor's diagonal argument Cantor's paradox Cantor's theorem Cardinality Cartesian closed category Cartesian product Categorial grammar Categorical theory Category (mathematics)Category of sets Category theory Church encoding Church–Turing thesis Class (set theory)Classical logic Codomain Combinatorial proof 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 Curry–Howard correspondence Cut-elimination theorem Dag Prawitz Data structures David Hilbert Decidability (logic)Decision problem Deductive system Diagonal lemma Diagram (mathematical logic)Dialectica interpretation Domain of a function Element (mathematics)Elementary diagram Elementary equivalence Elementary function arithmetic Elimination rule Empty set Encyclopedia of Mathematics Enumeration Equiconsistency Equivalence relation Euclid's Elements Euclidean geometry European Mathematical Society Existential quantification Expression (mathematics)Extension by definitions Extension by new constant and function names Extensionality Finitary Finitary relation Finite-valued logic Finite model theory Finite set First-order logic Fixed-point logic Focused proof 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 Fuzzy set General set theory Gerhard Gentzen Giuseppe Peano Gottlob Frege Grothendieck universe Ground expression Ground formula Grundlagen der Mathematik Gödel's completeness theorem Gödel's incompleteness theorems Gödel's second incompleteness theorem Gödel numbering Halting problem Hao Wang (academic)Harvey Friedman (mathematician)Hereditary set Higher-order logic Hilbert's axioms Hilbert's program 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 Interactive theorem proving Intermediate logic Interpretability logic Interpretation (logic)Interpretation (model theory)Interpretation function Intersection (set theory)Introduction rule Intuitionistic logic Intuitionistic type theory Isomorphism J. Barkley Rosser Jan Łukasiewicz Japaridze's polymodal logic Jean-Yves Girard Jon Barwise Kolmogorov complexity Kripke's theory of truth Kripke–Platek set theory Kurt Gödel Lambda calculus Large cardinal Lemma (mathematics)Lindström's theorem Linear logic Linguistics List (computer science)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 harmony Logical truth Logicism Löb's theorem Löwenheim–Skolem theorem Many-valued logic Map (mathematics)Martin Hugo Löb Material conditional Mathematical logic Mathematical object Mathematical proof Metalanguage Minimal axioms for Boolean algebra Modal logic Model complete theory Model theory Monadic predicate calculus Monadic second-order logic Montague grammar Morse–Kelley set theory NP (complexity)Naive set theory Natural deduction Natural deduction calculus Natural language semantics 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 Arithmetic Peano arithmetic Peano axioms Peer review Per Martin-Löf Philosophy of mathematics Power set Predicate (mathematical logic)Predicate logic Predicate variable Prime model Primitive recursive arithmetic Primitive recursive function Principia Mathematica Proof-theoretic semantics Proof (truth)Proof calculi Proof complexity Proof mining Proof net Proof of impossibility Proof techniques Proposition Propositional calculus Propositional formula Propositional logic Propositional variable Provability logic Quantifier (logic)Quantifier rank Real number Recursion Recursion theory Recursive data type Recursive set Reductive logic Relation (mathematics)Relevance logic 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 (mathematics)Set theory Signature (logic)Singleton (mathematics)Skolem arithmetic Solomon Feferman Soundness Spectrum of a sentence Spectrum of a theory Square of opposition Stanford Encyclopedia of Philosophy Stanisław Jaśkowski Strength (mathematical logic)String (formal languages)Structural proof theory Structure (mathematical logic)Substitution (logic)Substructural logic Substructure (mathematics)Supertask Supremum Surjective function Syllogism Symbol (formal)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 Theorem Theoretical computer science Theories of truth Theory (mathematical logic)Three-valued logic Timeline of mathematical logic Transfer principle Transfinite induction Transitive set Tree (data structure)True arithmetic Truth predicate Truth table Truth value Turing machine Type-logical grammar Type (model theory)Type theory Typed lambda calculus Ultrafilter (set theory)Ultraproduct Uncountable set Undecidable problem Uninterpreted function Union (set theory)Uniqueness quantification Universal quantification Universal set Universe (mathematics)Urelement Validity (logic)Van Nostrand (publisher)Variable (mathematics)Venn diagram Von Neumann universe Von Neumann–Bernays–Gödel set theory Well-formed formula ZF set theory Zermelo–Fraenkel set theory Zorn's lemma Ω-consistent theory

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