For Quine's theory sometimes called "Mathematical Logic", see New Foundations.For other uses, see Logic (disambiguation).
Part of a series on
Mathematics
History
Outline
Index
Areas
Number theory
Geometry
Algebra
Calculus and Analysis
Discrete mathematics
Logic and Set theory
Probability
Statistics and Decision theory
Relationship with sciences
Physics
Chemistry
Geosciences
Computation
Biology
Linguistics
Economics
Philosophy
Education
Mathematics Portal
v
t
e
Mathematical logic is the study of formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory (also known as computability theory). Research in mathematical logic commonly addresses the mathematical properties of formal systems of logic such as their expressive or deductive power. However, it can also include uses of logic to characterize correct mathematical reasoning or to establish foundations of mathematics.
Since its inception, mathematical logic has both contributed to and been motivated by the study of foundations of mathematics. This study began in the late 19th century with the development of axiomatic frameworks for geometry, arithmetic, and analysis. In the early 20th century it was shaped by David Hilbert's program to prove the consistency of foundational theories. Results of Kurt Gödel, Gerhard Gentzen, and others provided partial resolution to the program, and clarified the issues involved in proving consistency. Work in set theory showed that almost all ordinary mathematics can be formalized in terms of sets, although there are some theorems that cannot be proven in common axiom systems for set theory. Contemporary work in the foundations of mathematics often focuses on establishing which parts of mathematics can be formalized in particular formal systems (as in reverse mathematics) rather than trying to find theories in which all of mathematics can be developed.
and 19 Related for: Mathematical logic information
Mathematicallogic is the study of formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory...
addresses the mathematical properties of formal systems of logic. However, it can also include attempts to use logic to analyze mathematical reasoning or...
portal Glossary of logic Józef Maria Bocheński List of notation used in Principia Mathematica List of mathematical symbols Logic alphabet, a suggested...
20th century or had not previously been considered as mathematics, such as mathematicallogic and foundations. Number theory began with the manipulation...
This is the sense used in traditional Aristotelian logic, although in contemporary mathematicallogic the term satisfiable is used instead. The syntactic...
proof used in mathematics, a hearkening back to the Greek tradition. The development of the modern "symbolic" or "mathematical" logic during this period...
validate and discover new mathematical theorems and proofs. There has always been a strong influence from mathematicallogic on the field of artificial...
In mathematicallogic, New Foundations (NF) is an axiomatic set theory, conceived by Willard Van Orman Quine as a simplification of the theory of types...
A mathematical proof is a deductive argument for a mathematical statement, showing that the stated assumptions logically guarantee the conclusion. The...
power. In logic and the foundations of mathematics, formal languages are used to represent the syntax of axiomatic systems, and mathematical formalism...
foundations of mathematics started at the end of the 19th century and formed a new mathematical discipline called mathematicallogic, which later had...
Entscheidungsproblem" Introduction to Mathematical Philosophy "New Foundations for MathematicalLogic" Principia Mathematica The Simplest Mathematics History and philosophy...
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...
Modern mathematics formalizes its foundations to such an extent that mathematical theories can be regarded as mathematical objects, and mathematics itself...
false premises. Soundness has a related meaning in mathematicallogic, wherein a formal system of logic is sound if and only if every well-formed formula...
Discrete mathematics is the study of mathematical structures that can be considered "discrete" (in a way analogous to discrete variables, having a bijection...
used in mathematicallogic and computer science. Mathematical induction in this extended sense is closely related to recursion. Mathematical induction...
This is a list of mathematicallogic topics. For traditional syllogistic logic, see the list of topics in logic. See also the list of computability and...
timeline of mathematicallogic; see also history of logic. 1847 – George Boole proposes symbolic logic in The Mathematical Analysis of Logic, defining what...
Global Information
Mathematics Portal