Theorem

Definition

In mathematics and formal logic, a theorem is a statement that has been proven, or can be proven. The proof of a theorem is a logical argument that uses the inference rules of a deductive system to establish that the theorem is a logical consequence of the axioms and previously proved theorems.

Related concepts

Abstract logic Ackermann set theory Aesthetics of mathematics Aleph number Alfréd Rényi Algebraic logic Alphabet (formal languages)Antecedent (logic)Aphorism Archimedes Argument Arithmetic Arity Atomic formula Atomic model (mathematical logic)Atomic sentence Automata theory Automated theorem proving Axiom Axiom of choice Axiom schema Axiomatic system Axiomatization of Boolean algebras Banach–Tarski paradox Bayes' rule Belief Benson Mates Bijection Binary operation Boolean algebra Boolean algebras canonically defined Boolean function Bézout's identity 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 Classification of finite simple groups Codomain Collatz conjecture Combinatorics Compactness theorem Complement (set theory)Complete theory Completeness (logic)Computability theory Computable function Computable set Computably enumerable set Concrete category Conjecture Consequent Conservative extension Consistency Constructible universe Construction of the real numbers Constructive set theory Continuum hypothesis Contradiction Corollary Countable set Craig's theorem Cramer's rule Cut-elimination theorem Decidability (logic)Decision problem Deductive Deductive system Definition Diagram (mathematical logic)Domain of a function Doron Zeilberger Education Resources Information Center Element (mathematics)Elementary diagram Elementary equivalence Elementary function arithmetic Empty set Entscheidungsproblem Enumeration Epistemology Equiconsistency Equivalence relation Erdős number Eric W. 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Burgess Kepler conjecture Kolmogorov's zero–one law Kolmogorov complexity Kripke's theory of truth Kripke–Platek set theory Lambda calculus Large cardinal Law (mathematics)Law of cosines Law of large numbers Least-upper-bound principle Lemma (mathematics)Lindström's theorem Line (mathematics)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 List of theorems List of theorems called fundamental List of trigonometric identities Logic Logical argument Logical biconditional Logical conjunction Logical connective Logical consequence Logical constant Logical disjunction Logical equality Logical equivalence Logical truth Logically Logicism Löb's theorem Löwenheim–Skolem theorem Macro (computer science)Mandelbrot set Many-valued logic Map (mathematics)Material conditional MathWorld Mathematical logic Mathematical object Mathematical proof Mathematical rigor Mathematical theory Mathematics Merriam-Webster Mertens conjecture Metalanguage Metatheorem Minimal axioms for Boolean algebra Modal logic Model complete theory Model theory Monadic predicate calculus Monadic second-order logic Morse–Kelley set theory NP (complexity)Naive set theory Natural deduction Natural number Necessity and sufficiency Negation New Foundations Non-Euclidean geometries Non-Euclidean geometry Non-classical logic Non-logical symbol Non-standard model Non-standard model of arithmetic Nonsense Number theory Open formula Operation (mathematics)Ordinal analysis Ordinal number P (complexity)P versus NP problem Paradoxes of set theory Partition of a set Paul Erdős Paul Halmos Paul Hoffman (science writer)Peano arithmetic Peano axioms Philosophy of mathematics Physical world Pigeonhole principle Plane (mathematics)Poincaré conjecture Postulate Power set Predicate (mathematical logic)Predicate logic Predicate variable Premise Prime model Primitive recursive arithmetic Primitive recursive function Principia Mathematica Principle Probability distribution Probability theory Proof of impossibility Proof theory Proposition Propositional calculus Propositional formula Propositional logic Propositional variable Pythagorean theorem Q.E.D.Quantifier (logic)Quantifier rank Rectangle Recursion Recursive set Relation (mathematics)Reverse mathematics Riemann hypothesis Riemann zeta function Right angle Robinson arithmetic Rule of inference Russell's paradox Satisfiability Saturated model Schröder–Bernstein theorem Scientific law Second-order arithmetic Second-order logic Self-evidence Self-verifying theories Semantic theory of truth Semantics Semantics of logic Sentence (mathematical logic)Sequent calculus Set (mathematics)Set theory Signature (logic)Singleton (mathematics)Skolem arithmetic Soundness Special case Spectrum of a sentence Spectrum of a theory Square Square of opposition Statement (logic)Strength (mathematical logic)String (computer science)String (formal languages)Structure (mathematical logic)Style guide Substitution (logic)Substructure (mathematics)Supertask 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 The Man Who Loved Only Numbers Theories of truth Theory Theory (disambiguation)Theory (mathematical logic)Theory of justification Thomas Heath (classicist)Three-valued logic Timeline of mathematical logic Tombstone (typography)Toy theorem Transfer principle Transitive set Triangle Triviality (mathematics)True arithmetic Truth function Truth predicate Truth table Truth value Turing machine Type (model theory)Type theory U.S. Department of Education Ultrafilter (set theory)Ultraproduct Uncountable set Undecidable problem Uninterpreted function Union (set theory)Uniqueness quantification Universal quantification Universal set Universality (dynamical systems)Universe (mathematics)Urelement Validity (logic)Vandermonde's identity Variable (mathematics)Venn diagram Von Neumann universe Von Neumann–Bernays–Gödel set theory Well-formed formula Well-formed formulas Wiles's proof of Fermat's Last Theorem Wilfrid Hodges Zermelo–Fraenkel set theory Zorn's lemma

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