blanketglossary

Foundations of mathematics

Definition

Foundations of mathematics are the logical and mathematical frameworks that allow the development of mathematics without generating self-contradictory theories, and to have reliable concepts of theorems, proofs, algorithms, etc. in particular. This may also include the philosophical study of the relation of this framework with reality.

Related concepts

(ε, δ)-definition of limit Abraham Fraenkel Abstract algebra Abstract logic Ackermann set theory Actual infinity Affine space Al-Khwarizmi Alan Turing Aleph number Alfred North Whitehead Alfred Tarski Algebra Algebraic geometry Algebraic logic Algebraic number theory Algebraic operation Algebraic topology Algorithm Algorithmic information theory Alonzo Church Alphabet (formal languages)American Mathematical Monthly Analytic geometry Analytic number theory Ancient Greek philosophers Anne Sjerp Troelstra Antipodal point Applied mathematics Areas of mathematics Argument Aristotelian realist philosophy of mathematics Aristotle Aristotle's Arithmetic Arithmetic geometry Arithmetization of analysis Arity Astrology Astronomy Atomic formula Atomic model (mathematical logic)Atomic sentence Augustus De Morgan Automata theory Automated theorem proving Axiom Axiom of choice Axiom of reducibility Axiom schema Axiomatic method Axiomatic set theory Axiomatic system Axiomatic theory Axiomatization of Boolean algebras Banach–Tarski paradox Bernard Bolzano Bernhard Riemann Bertrand Russell Bijection Binary operation Boolean algebra Boolean algebras canonically defined Boolean function Bounded set Brouwer–Hilbert controversy Calculus Calculus of constructions Calculus of inductive constructions Cantor's diagonal argument Cantor's paradox Cantor's theorem Cardinal number Cardinality Carl Friedrich Gauss Cartesian coordinate system Cartesian product Categorical theory Category (mathematics)Category of sets Category theory Cauchy Cauchy sequence Cauchy sequences Charles Méray Charles Sanders Peirce Church encoding Church–Turing thesis Class (set theory)Classical logic Codomain Cognitive science of mathematics Combinatorics Commutative algebra Compactness theorem Complement (set theory)Complete theory Completeness (logic)Completeness of the real numbers Completeness theorem Complex analysis Complex geometry Computability theory Computable function Computable set Computably enumerable set Computational complexity theory Computational geometry Computational mathematics Computer algebra Computer science Conceptualism Concrete category Conservative extension Consistency Consistency proof Consistency strength Consistent theory Constructible universe Construction of the real numbers Constructive set theory Constructivism (mathematics)Continuous function Continuous functions Continuum (set theory)Continuum hypothesis Control theory Controversy over Cantor's theory Coordinates Countable set Cours d'Analyse Cross-ratio Cubic formula David E. Rowe David Hilbert De Morgan's laws Decidability (logic)Decision problem Decision theory Dedekind cut Deductive system Dependent type Derivative Descriptive set theory Determinacy Diagram (mathematical logic)Differential equation Differential geometry Differential topology Diophantine equation Diophantine geometry Discrete geometry Discrete mathematics Domain of a function Dover Publications Downward Löwenheim–Skolem theorem Edmund F. Robertson Eduard Heine Edward N. Zalta Effective method Element (mathematics)Elementary algebra Elementary diagram Elementary equivalence Elementary function arithmetic Ellipse Elliptic geometry Emil Artin Empty set Engineering mathematics Entscheidungsproblem Enumeration Epistemology Equiconsistency Equivalence relation Ernst Zermelo Errett Bishop Euclid Euclid's Elements Euclidean geometry Eudoxus of Cnidus Existential quantification Experiment Expression (mathematics)Extension by definitions Extension by new constant and function names Extensionality Field (mathematics)Finitary relation Finite-valued logic Finite geometry Finite model theory Finite set Finitely presented group Finitism First-order logic Fixed-point logic Forcing (mathematics)Formal grammar Formal language Formal logic Formal logical system Formal proof Formal semantics (logic)Formal system Formalism (mathematics)Formation rule Foundational crisis of mathematics Foundations of Mathematics (book)Foundations of geometry Free logic Free variables and bound variables Function (mathematics)Functional analysis Functional predicate Future of mathematics Fuzzy set Game theory General set theory General topology Generality of algebra Gentzen's consistency proof Geomathematics Geometric Algebra (book)Geometric topology Geometrical space Geometry Georg Cantor George Berkeley George Boole Girard's paradox Giuseppe Peano Glossary of category theory Glossary of mathematical symbols Gottfried Wilhelm Leibniz Gottlob Frege Graph theory Great circle Greek mathematics Greek philosophers Greek philosophy Gregory Chaitin Grothendieck universe Ground expression Ground formula Group theory Gödel's completeness theorem Gödel's incompleteness theorem Gödel's incompleteness theorems Gödel numbering Halting problem Harmonic analysis Henri Poincaré Hereditary set Hermann Hankel Hermann Weyl Higher-order logic Higher category theory Hilary Putnam Hilbert's axioms Hilbert's problems Hilbert's program Hilbert's tenth problem Hilbert system History of logic History of mathematical logic History of mathematics History of type theory Homological algebra Homotopy theory Homotopy type theory Howard Eves Hyperbolic functions Hyperbolic geometry Hyperbolic triangle Hypercomplex analysis Hypersphere Image (mathematics)Implementation of mathematics in set theory Inaccessible cardinal Inconsistency Inconsistent Independence (mathematical logic)Inference Inference rule Inference rules Infinite-valued logic Infinite sequence Infinite sequences Infinite series Infinite set Infinitesimal Infinitesimal calculus Infinity Informal mathematics Information theory Inhabited set Injective function Internet Encyclopedia of Philosophy Interpretation (logic)Interpretation (model theory)Interpretation function Intersection (set theory)Intuitionism Intuitionistic type theory Irrational number Isaac Newton Isomorphism Jeremy Avigad Jeremy Gray (mathematician)Joel David Hamkins Johann Heinrich Lambert János Bolyai Karl Weierstrass Karl von Staudt Knot theory Kolmogorov complexity Kripke's theory of truth Kripke–Platek set theory Kurt Gödel L. E. J. Brouwer La Géométrie Lambda calculus Large cardinal Large numbers Least upper bound Lemma (mathematics)Leopold Kronecker Leopold Löwenheim Liar paradox Limit (mathematics)Lindström's theorem Line (geometry)Linear algebra List of Hilbert systems List of axioms List of category theory topics List of films about mathematicians List of first-order theories List of formal systems List of mathematical logic topics List of mathematical theories List of mathematicians List of paradoxes List of set identities and relations List of set theory topics List of statements independent of ZFC Lists of mathematics topics Logic Logical biconditional Logical conjunction Logical connective Logical consequence Logical constant Logical disjunction Logical equality Logical equivalence Logical truth Logicism Luitzen Egbertus Jan Brouwer Löwenheim–Skolem theorem MacTutor History of Mathematics Archive Many-valued logic Map (mathematics)Material conditional Mathematical analysis Mathematical and theoretical biology Mathematical chemistry Mathematical constructivism Mathematical economics Mathematical finance Mathematical geology Mathematical induction Mathematical infinity Mathematical intuition Mathematical linguistics Mathematical logic Mathematical object Mathematical optimization Mathematical physics Mathematical practice Mathematical proof Mathematical psychology Mathematical sociology Mathematical statistics Mathematical truth Mathematicians Mathematics Mathematics and art Mathematics education Measure (mathematics)Metalanguage Metalogic Metamathematics Metatheorem Metatheory Middle Ages Minimal axioms for Boolean algebra Modal logic Model complete theory Model theory Monadic predicate calculus Monadic second-order logic Morse–Kelley set theory Multilinear algebra Multiverse NP (complexity)Naive set theory Natural deduction Natural number Negation New Foundations Newton's law of gravitation Nicolas Bourbaki Nikolai Lobachevsky Nominalism Non-Euclidean geometry Non-logical symbol Non-standard model Non-standard model of arithmetic Number Number theory Numerical analysis Ontology Open formula Operation (mathematics)Operations research Orbit Order theory Ordinal analysis Ordinal number Ordinal numeral P (complexity)P versus NP problem Pappus hexagon theorem Paradox Paradoxes of set theory Paradoxical Parallel postulate Partition of a set Paul Cohen (mathematician)Peano arithmetic Peano axioms Per Martin-Löf Philosophical realism Philosophy of mathematics Plane (geometry)Plato Platonism Point (geometry)Posterior Analytics Postulate Potential infinity Power set Predicate (mathematical logic)Predicate logic Predicate variable Premise Prime model Primitive recursive arithmetic Primitive recursive function Principia Mathematica Probability Probability theory Projective geometry Projective space Proof (mathematics)Proof assistant Proof of impossibility Proof theory Proposition Propositional calculus Propositional formula Propositional variable Pyotr Novikov Pythagoreanism Quantification (logic)Quantifier (logic)Quantifier rank Quartic formula Quasi-empiricism in mathematics Quine–Putnam indispensability argument Quine–Putnam indispensability thesis Rational number Raymond Louis Wilder Real analysis Real number Reality Recreational mathematics Recursion Recursive set Reductio ad absurdum Relation (mathematics)René Descartes Reuben Hersh Reverse Mathematics Reverse mathematics Richard Dedekind Richard Feynman Robinson arithmetic Rule of inference Russel's paradox Russell's paradox Sanskrit prosody Satisfiability Saturated model Schröder–Bernstein theorem Second-order arithmetic Second-order logic Self-verifying theories Semantic theory of truth Semantics of logic Semi-decidable Sentence (mathematical logic)Sequent calculus Set-theoretic Platonism Set (mathematics)Set theory Signature (logic)Singleton (mathematics)Skolem's paradox Skolem arithmetic Social group Soundness Spectrum of a sentence Spectrum of a theory Springer Science+Business Media Square of opposition Stanford Encyclopedia of Philosophy Statistics Stephen Kleene Steven Weinberg Strength (mathematical logic)String (formal languages)Structuralism (philosophy of mathematics)Structure (mathematical logic)Substitution (logic)Substructure (mathematics)Successor function Supertask Surjective function Surveying Suslin's problem Syllogism Syllogisms Syllogistic logic Symbol (formal)Syntax (logic)Synthetic geometry Systems science 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 Analyst Theorem Theories of truth Theory Theory (mathematical logic)Theory of computation Thomas Tymoczko Thoralf Skolem Three-valued logic Timeline of mathematical logic Timeline of mathematics Topology Topos Transfer principle Transitive set True arithmetic Truth (mathematics)Truth predicate Truth table Truth value Turing machine Type (model theory)Type theory Type–token distinction Ultrafilter (set theory)Ultraproduct Uncountable set Undecidable problem Uninterpreted function Union (set theory)Uniqueness quantification Univalent foundations Universal algebra Universal quantification Universal set Universe (mathematics)University of St Andrews Urelement Urelements Use–mention distinction Validity (logic)Variable (mathematics)Vector space Venn diagram Von Neumann universe Von Neumann–Bernays–Gödel set theory Weierstrass function Well-formed formula Wilbur Dyre Hart Willard Quine Word problem for groups ZFC Zeno of Elea Zermelo set theory Zermelo–Fraenkel set theory ∞-groupoid ∞-topos

172 concepts already in your glossary