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Set of sentences in a formal language
In mathematical logic, a theory (also called a formal theory) is a set of sentences in a formal language. In most scenarios a deductive system is first
Theory_(mathematical_logic)
Subfield of mathematics
Mathematical logic is the study of formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory
Mathematical_logic
Type of logical system
Mathematical Logic, John Wiley and Sons, 1967. F. R. Drake, Set theory: An introduction to large cardinals (1974) Rogers, R. L., Mathematical Logic and
First-order_logic
discrete and Euclidean geometries, graph theory, group theory, mathematical logic, number theory, set theory, Ramsey theory, dynamical systems, and partial differential
List of unsolved problems in mathematics
List_of_unsolved_problems_in_mathematics
metamathematics—the study of mathematics using mathematical methods to produce metatheories, or mathematical theories about other mathematical theories. Early investigations
History_of_logic
Basic framework of mathematics
of mathematics. The resolution of this crisis involved the rise of a new mathematical discipline called mathematical logic that includes set theory, model
Foundations_of_mathematics
This is a list of mathematical logic topics. For traditional syllogistic logic, see the list of topics in logic. See also the list of computability and
List of mathematical logic topics
List_of_mathematical_logic_topics
Non-contradiction of a theory
In deductive logic, a consistent theory is one that does not lead to a logical contradiction. A theory T {\displaystyle T} is consistent if there is no
Consistency
Formal system in mathematical logic
In mathematical logic, an abstract logic is a formal system consisting of a class of sentences and a satisfaction relation with specific properties related
Abstract_logic
In mathematics, a statement that has been proven
important theorems. In mathematical logic, the concepts of theorems and proofs have been formalized in order to allow mathematical reasoning about them
Theorem
Mathematical theory of data types
In mathematical logic, and theoretical computer science, type theory is the study of formal systems that classify expressions or mathematical objects by
Type_theory
Term in mathematical logic
In mathematical logic, independence is the unprovability of some specific sentence from some specific set of other sentences. The sentences in this set
Independence (mathematical logic)
Independence_(mathematical_logic)
Branch of mathematical logic
Proof theory is a major branch of mathematical logic and theoretical computer science within which proofs are treated as formal mathematical objects, facilitating
Proof_theory
In mathematical logic, a well-formed formula with no free variables
In mathematical logic, a sentence (or closed formula) of a predicate logic is a Boolean-valued well-formed formula with no free variables. A sentence
Sentence_(mathematical_logic)
Symbol representing a property or relation in logic
Andreevich; Maksimova, Larisa (2003). Problems in Set Theory, Mathematical Logic, and the Theory of Algorithms. New York: Springer. p. 52. ISBN 0306477122
Predicate_(logic)
Polish–American mathematician (1901–1983)
model theory, metamathematics, and algebraic logic, he also contributed to abstract algebra, topology, geometry, measure theory, mathematical logic, set
Alfred_Tarski
Area of mathematical logic
In mathematical logic, model theory is the study of the relationship between formal theories (a collection of sentences in a formal language expressing
Model_theory
Academic subfield of computer science
with that property. Computability theory is closely related to the branch of mathematical logic called recursion theory, which removes the restriction of
Theory_of_computation
Description of non-logical symbols
In mathematical logic, a signature is a description of the non-logical symbols of a formal language. In universal algebra, a signature lists the operations
Signature_(logic)
Axiomatic set theory devised by W.V.O. Quine
In mathematical logic, New Foundations (NF) is a non-well-founded, finitely axiomatizable set theory conceived by Willard Van Orman Quine as a simplification
New_Foundations
Basic notion of sameness in mathematics
of symbolic logic. There are generally two ways that equality is formalized in mathematics: through logic or through set theory. In logic, equality is
Equality_(mathematics)
Reasoning about equations with free variables
In mathematical logic, algebraic logic is the reasoning obtained by manipulating equations with free variables. What is now usually called classical algebraic
Algebraic_logic
Type of theory in mathematical logic
In mathematical logic, a theory is categorical if it has exactly one model (up to isomorphism). Such a theory can be viewed as defining its model, uniquely
Categorical_theory
even formal theories are considered as mathematical objects in proof theory. In philosophy of mathematics, the concept of "mathematical objects" touches
Mathematical_object
timeline of mathematical logic; see also history of logic. 1847 – George Boole proposes symbolic logic in The Mathematical Analysis of Logic, defining what
Timeline of mathematical logic
Timeline_of_mathematical_logic
In model theory, a subfield of mathematical logic, an atomic model is a model such that the complete type of every tuple is axiomatized by a single formula
Atomic model (mathematical logic)
Atomic_model_(mathematical_logic)
List of symbols used to express logical relations
portal Glossary of logic Józef Maria Bocheński List of notation used in Principia Mathematica List of mathematical symbols Logic alphabet, a suggested
List_of_logic_symbols
Study of computable functions and Turing degrees
Computability theory, also known as recursion theory, is a branch of mathematical logic, computer science, and the theory of computation that originated
Computability_theory
Reasoning for mathematical statements
A mathematical proof is a deductive argument for a mathematical statement, showing that the stated assumptions logically guarantee the conclusion. The
Mathematical_proof
Study of discrete mathematical structures
Discrete mathematics is the study of mathematical structures that can be considered "discrete" (in a way analogous to discrete variables, having a one-to-one
Discrete_mathematics
Index of articles associated with the same name
Stratification has several usages in mathematics. In mathematical logic, stratification is any consistent assignment of numbers to predicate symbols guaranteeing
Stratification_(mathematics)
Whether a decision problem has an effective method to derive the answer
determined. Zeroth-order logic (propositional logic) is decidable, whereas first-order and higher-order logic are not. A theory (set of sentences closed
Decidability_(logic)
Branch of mathematics that studies sets
Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any
Set_theory
American philosopher
decision theory, mathematical logic and philosophy of mathematics.[citation needed] Easwaran is an associate editor of Journal of Philosophical Logic. Two
Kenny_Easwaran
Limitative results in mathematical logic
theorems are two theorems of mathematical logic that are concerned with the limits of provability in formal axiomatic theories. These results, published
Gödel's incompleteness theorems
Gödel's_incompleteness_theorems
Standard system of axiomatic set theory
Peter (2005). Fundamentals of Mathematical Logic. A K Peters. ISBN 978-1-56881-262-5. Jech, Thomas (2003). Set Theory: The Third Millennium Edition,
Zermelo–Fraenkel_set_theory
Branch of logic using category theory to study mathematical structures
Categorical logic is the branch of mathematics in which tools and concepts from category theory are applied to the study of mathematical logic. It is also
Categorical_logic
Study of the semantics, or interpretations, of formal and natural languages
functions from them to some predefined mathematical domains: an interpretation of first-order predicate logic is given by a mapping from terms to a universe
Semantics_(logic)
School of thought in philosophy of mathematics
is an extension of logic, some or all of mathematics is reducible to logic, or some or all of mathematics may be modelled in logic. Bertrand Russell and
Logicism
Sequence of words formed by specific rules
power. In logic and the foundations of mathematics, formal languages are used to represent the syntax of axiomatic systems, and mathematical formalism
Formal_language
Alternative foundation of mathematics
core design of constructive logic using dependent types. Martin-Löf designed the type theory on the principles of mathematical constructivism. Constructivism
Intuitionistic_type_theory
Mapping of mathematical formulas to a particular meaning
(although related) meaning in model theory; see interpretation (model theory). In the context of mathematical logic, the term "model" was first used in
Structure (mathematical logic)
Structure_(mathematical_logic)
Entscheidungsproblem" Introduction to Mathematical Philosophy "New Foundations for Mathematical Logic" Principia Mathematica The Simplest Mathematics History and philosophy
Philosophy_of_mathematics
Glossary of graph theory List of graph theory topics Logic is the foundation that underlies mathematical logic and the rest of mathematics. It tries to formalize
Lists_of_mathematics_topics
Mathematical logic concept
In mathematical logic and philosophy, Skolem's paradox is the apparent contradiction that a countable model of first-order set theory could contain an
Skolem's_paradox
Form of entertainment in mathematics
explicit mathematics in order to play mathematical games. For example, Mancala is studied in the mathematical field of combinatorial game theory, but no
Recreational_mathematics
Type theory in logic and mathematics
In mathematical logic and computer science, homotopy type theory (HoTT) includes various lines of development of intuitionistic type theory, based on the
Homotopy_type_theory
Field of knowledge
relationship between mathematical truth, logic, and reality is a subject of philosophical debate. Some areas of mathematics, such as game theory, are developed
Mathematics
compactness theorem (mathematical logic) Borel determinacy theorem (set theory) Büchi-Elgot-Trakhtenbrot theorem (mathematical logic) Cantor–Bernstein–Schröder
List_of_theorems
Mathematician
in set theory, mathematical logic, and infinitary combinatorics. She is a privatdozent in the Kurt Gödel Research Center for Mathematical Logic at the
Vera_Fischer_(mathematician)
Existence and cardinality of models of logical theories
In mathematical logic, the Löwenheim–Skolem theorem is a theorem on the existence and cardinality of models, named after Leopold Löwenheim and Thoralf
Löwenheim–Skolem_theorem
Study of Lie groups, Lie algebras and differential equations
is expressed by root systems and root data. Lie theory has been particularly useful in mathematical physics since it describes the standard transformation
Lie_theory
Area of mathematics
systems, mathematical dynamics, mathematical dynamical systems theory or the mathematical theory of dynamical systems. Dynamical systems theory and chaos
Dynamical_systems_theory
Theory of logic to account for observations from quantum theory
In the mathematical study of logic and the physical analysis of quantum foundations, quantum logic is a set of rules for manipulation of propositions
Quantum_logic
Book by Wilhelm Ackermann
formalism of mathematical logic, and is presupposed by contemporary treatments of Peano arithmetic and nearly all treatments of axiomatic set theory. The 1928
Principles of Mathematical Logic
Principles_of_Mathematical_Logic
System of formal mathematical logic
foundation for mathematics comparable to first-order logic plus set theory. It is a form of higher-order logic and closely related to the logics of the HOL
Q0_(mathematical_logic)
Syntactically correct logical formula
In mathematical logic, propositional logic, and predicate logic, a well-formed formula, abbreviated WFF or wff, often simply formula, is a finite sequence
Well-formed_formula
Statement in a metalanguage
judgment in mathematical logic can be exploited also in foundation of type theory as well. Simply typed lambda calculus Mathematical logic Martin-Löf,
Judgment_(mathematical_logic)
Form of second-order logic
In mathematical logic, monadic second-order logic (MSO) is the fragment of second-order logic where the second-order quantification is limited to quantification
Monadic_second-order_logic
Mathematical term; concerning axioms used to derive theorems
other statements. In mathematics these logical consequences of the axioms may be known as lemmas or theorems. A mathematical theory is an expression used
Axiomatic_system
Book by George Boole
the Mathematical Theories of Logic and Probabilities by George Boole, published in 1854, is the second of Boole's two monographs on algebraic logic. Boole
The_Laws_of_Thought
Concept in mathematical logic
In mathematical logic, a theory of a language is complete if it is consistent and it proves every closed formula with which it is not inconsistent. That
Complete_theory
Branch of mathematics
Order theory is a branch of mathematics that investigates the intuitive notion of order using binary relations. It provides a formal framework for describing
Order_theory
Formal system of logic
In mathematics and logic, a higher-order logic (HOL) is a form of logic that is distinguished from first-order logic by additional quantifiers and, sometimes
Higher-order_logic
System for reasoning about vagueness
identical at first, but fuzzy logic uses degrees of truth as a mathematical model of vagueness, while probability is a mathematical model of ignorance. A basic
Fuzzy_logic
American-Turkish mathematician (born 1992)
American Mathematical Society. December 9, 2014. Retrieved July 23, 2026 – via EurekAlert!. Cepelewicz, Jordana (November 29, 2022). "Mathematical Trio Advances
Levent_Alpöge
Characteristic of some logical systems
In mathematical logic and metalogic, a formal system is called complete with respect to a particular property if every formula having the property can
Completeness_(logic)
Topics referred to by the same term
The term mathematical theory may refer to: Theory (mathematical logic), a collection of sentences in a formal language. Mathematical theory, a branch of
Mathematical theory (disambiguation)
Mathematical_theory_(disambiguation)
Mathematics independent of applications
new mathematical objects or working out the mathematical consequences of basic principles. While the distinction between pure and applied mathematics has
Pure_mathematics
Geometric logic is capable of expressing many mathematical theories and has close connections to topos theory. A theory of first-order logic is geometric
Geometric_logic
Many-valued logic in which truth values comprise a continuous range
also led to generalizations of mathematical theories in the family of t-norm fuzzy logics. Basic fuzzy logic is the logic of continuous t-norms (binary
Infinite-valued_logic
Academic discipline
validate and discover new mathematical theorems and proofs. There has always been a strong influence from mathematical logic on the field of artificial
Logic_in_computer_science
Use of logic to perform or reason about computation
logic is the use of logic to perform or reason about computation. It bears a similar relationship to computer science and engineering as mathematical
Computational_logic
Class of formal logics
logic) didn't meet the requirements to be a logic, saying that it was "set theory in disguise". Classical logic is the standard logic of mathematics.
Classical_logic
3-volume treatise on mathematics, 1910–1913
methods of mathematical logic and to minimise the number of primitive notions, axioms, and inference rules; to precisely express mathematical propositions
Principia_Mathematica
Type of infinite structure
In mathematical logic, and more specifically in model theory, an infinite structure ( M , < , … ) {\displaystyle (M,<,\dots )} that is totally ordered
O-minimal_theory
Decidable theory of equality
In mathematical logic the theory of pure equality is a first-order theory. It has a signature consisting of only the equality relation symbol, and includes
Theory_of_pure_equality
Overview of and topical guide to discrete mathematics
Graph theory – Area of discrete mathematics a study of graphs – Vertices connected in pairs by edges Mathematical logic – Subfield of mathematics Discrete
Outline of discrete mathematics
Outline_of_discrete_mathematics
General theory of mathematical structures
algebraic topology. Category theory can be used in most areas of mathematics. In particular, many constructions of new mathematical objects from previous ones
Category_theory
Method of deriving conclusions
proof by contradiction, and mathematical induction. Mathematical logic, a subfield of mathematics and logic, uses mathematical methods and frameworks to
Rule_of_inference
Mathematical Logic as based on the theory of types in van Heijenoort 1967:150 cf. commentary by W. V. O. Quine before Russell's (1908) Mathematical logic
History_of_type_theory
Rules used for constructing, or transforming the symbols and words of a language
composition of well-formed expressions in a programming language. As in mathematical logic, it is independent of semantics and interpretation. A symbol is an
Syntax_(logic)
Branch of mathematical logic
Reverse mathematics is a program in mathematical logic that seeks to determine which axioms are required to prove theorems of mathematics. Its defining
Reverse_mathematics
Overview of and topical guide to logic
Ternary relation Transitive relation Trichotomy (mathematics) Well-founded relation Mathematical logic Set theory list Aleph null Bijection, injection and surjection
Outline_of_logic
Polish mathematician (1920–1998)
He is known for his contributions to model theory, mathematical logic, Abelian group theory and mathematical economics. Łoś was born on 22 March 1920 in
Jerzy_Łoś
Concept in model theory
systems of formal logic can be defined via model theory. Specifically, a logic α {\displaystyle \alpha } is said to be as strong as a logic β {\displaystyle
Strength_(mathematical_logic)
Concept in model theory
In model theory, a branch of mathematical logic, the diagram of a structure is the set of sentences with parameters from the structure that are true in
Diagram_(mathematical_logic)
About mathematical infinity
In mathematical logic, the theory of infinite sets was first developed by Georg Cantor. Although this work has become a thoroughly standard fixture of
Controversy over Cantor's theory
Controversy_over_Cantor's_theory
branch of mathematics that studies mathematical systems and theories from a higher-level perspective, often using methods from mathematical logic. metatheorem
Glossary_of_logic
the applications of formal logic to mathematics. Mathematical optimization Mathematical physics The development of mathematical methods suitable for application
Glossary of areas of mathematics
Glossary_of_areas_of_mathematics
Branch of logic
first-order logic (FO). These invalidities all follow from Trakhtenbrot's theorem. While model theory has many applications to mathematical algebra, finite
Finite_model_theory
Statement that is taken to be true
speaking, a metaproof. These examples are metatheorems of our theory of mathematical logic since we are dealing with the very concept of proof itself. Aside
Axiom
Mathematical logic theory with exactly one countably infinite model up to isomorphism
In mathematical logic, an omega-categorical theory is a theory that has exactly one countably infinite model up to isomorphism. Omega-categoricity is
Omega-categorical_theory
in: type theory, constructive mathematics. Founder(s): K F. Gödel (1933) showed that intuitionistic logic can be embedded into modal logic S4. (other
Constructive_logic
Logical formulation of recursion
In mathematical logic, fixed-point logics are extensions of classical predicate logic that have been introduced to express recursion. Their development
Fixed-point_logic
mathematical logic, abstract model theory is a generalization of model theory that studies the general properties of extensions of first-order logic and
Abstract_model_theory
the concept in mathematical logic. For the concepts in sociology, see Institutional theory and Institutional logic. In mathematical logic, institutional
Institutional_model_theory
Algebraic manipulation of "true" and "false"
In mathematics and mathematical logic, Boolean algebra is a branch of algebra. It differs from elementary algebra in two ways. First, the values of the
Boolean_algebra
Form of logic that allows quantification over predicates
propositional logic. Second-order logic is in turn extended by higher-order logic and type theory. First-order logic quantifies only variables that range
Second-order_logic
1956 computer program written by Allen Newell, Herbert A. Simon and Cliff Shaw
proof in The Journal of Symbolic Logic, but it was rejected on the grounds that a new proof of an elementary mathematical theorem was not notable, apparently
Logic_Theorist
THEORY MATHEMATICAL-LOGIC
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THEORY MATHEMATICAL-LOGIC
THEORY MATHEMATICAL-LOGIC
THEORY MATHEMATICAL-LOGIC
THEORY MATHEMATICAL-LOGIC
THEORY MATHEMATICAL-LOGIC
THEORY MATHEMATICAL-LOGIC
THEORY MATHEMATICAL-LOGIC