Formal proof

Definition

In logic and mathematics, a formal proof or derivation is a finite sequence of sentences, each of which is an axiom, an assumption, or follows from the preceding sentences in the sequence, according to the rule of inference. It differs from a natural language argument in that it is rigorous, unambiguous and mechanically verifiable. If the set of assumptions is empty, then the last sentence in a formal proof is called a theorem of the formal system. The notion of theorem is generally effective, but there may be no method by which we can reliably find proof of a given sentence or determine that none exists. The concepts of Fitch-style proof, sequent calculus and natural deduction are generalizations of the concept of proof.

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Abstract logic Ackermann set theory Aleph number Algebraic logic Alphabet Alphabet (formal languages)Argument Arity Atomic formula Atomic model (mathematical logic)Atomic sentence Automata theory Automated theorem prover Automated theorem proving Axiom Axiom of choice Axiom schema Axiomatic system Axiomatization of Boolean algebras Banach–Tarski paradox 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 Codomain Compactness theorem Complement (set theory)Complete theory Computability theory Computable function Computable set Computably enumerable set Computationally intractable Concrete category Conservative extension Consistency Constructible universe Construction of the real numbers Constructive set theory Continuum hypothesis Contradiction Countable set De Bruijn Factor Decidability (logic)Decision problem Deductive apparatus Deductive system Diagram (mathematical logic)Domain of a function Effective method Element (mathematics)Elementary diagram Elementary equivalence Elementary function arithmetic Empty set Enumeration Equiconsistency Equivalence relation Euclid's Elements Euclidean geometry Existential quantification Expression (mathematics)Extension by definitions Extension by new constant and function names Extensionality False (logic)Finitary relation Finite-valued logic Finite model theory Finite set First-order logic Fitch notation Fixed-point logic Follows from Forcing (mathematics)Formal grammar Formal language Formal semantics (logic)Formal system Formal verification Formation rule Foundations of geometry Foundations of mathematics Free logic Free variables and bound variables Function (mathematics)Functional predicate Fuzzy set General set theory Generalization Grothendieck universe Ground expression Ground formula Gödel's completeness theorem Gödel's incompleteness theorems Gödel numbering Halting problem 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 Interactive theorem proving Interpretation (logic)Interpretation (model theory)Interpretation function Intersection (set theory)Isomorphism 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 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)Material conditional Mathematical logic Mathematical object Mathematical proof Mathematics Meaning (linguistics)Metalanguage Minimal axioms for Boolean algebra 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 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 Power set Predicate (mathematical logic)Predicate logic Predicate variable Prime model Primitive recursive arithmetic Primitive recursive function Principia Mathematica Proof (truth)Proof assistant Proof calculus Proof of impossibility Proof theory Proposition Proposition (philosophy)Propositional calculus Propositional formula Propositional variable Quantifier (logic)Quantifier rank Recursion Recursive set Reference Relation (mathematics)Reverse mathematics 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 Semantics of logic Semi-decidable Sentence (mathematical logic)Sequence Sequence (mathematics)Sequent calculus Set (mathematics)Set theory Signature (logic)Singleton (mathematics)Skolem arithmetic Soundness Spectrum of a sentence Spectrum of a theory Square of opposition Strength (mathematical logic)String (computer science)String (formal languages)Structure (mathematical logic)Substitution (logic)Substructure (mathematics)Supertask Surjective function Syllogism Symbol Symbol (formal)Syntax (logic)T-schema Tarski's axiomatization of the 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