Mathematical Logic

Mathematical logic (also known as symbolic logic) is a subfield of mathematics with close connections to the foundations of mathematics, theoretical computer science and philosophical logic. The field includes both the mathematical study of logic and the applications of formal logic to other areas of mathematics. The unifying themes in mathematical logic include the study of the expressive power of formal systems and the deductive power of formal proof systems.

Mathematical logic is often divided into the fields of set theory, model theory, recursion theory, and proof theory. These areas share basic results on logic, particularly first-order logic, and definability. In computer science (particularly in the ACM Classification) mathematical logic encompasses additional topics not detailed in this article; see logic in computer science for those.

Since its inception, mathematical logic has both contributed to, and has 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.

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Alex Wilkie
1948) is a British mathematician known for his contributions to Model theory and logic ... Previously Reader in Mathematical Logic at the University of Oxford, he was appointed to the Fielden Chair of Pure Mathematics at the University of Manchester in 2007 ... in mathematics with first class honours from University College London in 1969, his MSc (in mathematical logic) from the University of London in 1970, and his PhD ...
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Modal Operator - Modality Interpreted - Doxastic
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... In order to clarify the foundations of mathematics, the fields of mathematical logic and set theory were developed ... Mathematical logic includes the mathematical study of logic and the applications of formal logic to other areas of mathematics set theory is the branch of mathematics that studies sets or ... Category theory, which deals in an abstract way with mathematical structures and relationships between them, is still in development ...
Mathematical Logic - Foundations of Mathematics
... forward by constructivists in the 20th century, the mathematical community as a whole rejected them ... that Cantor has created." With the development of formal logic, Hilbert asked whether it would be possible to prove that an axiom system is consistent by analyzing the structure of possible proofs in the system ... the history of foundations of mathematics involves nonclassical logics and constructive mathematics ...

Famous quotes containing the words logic and/or mathematical:

The much vaunted male logic isn’t logical, because they display prejudices—against half the human race—that are considered prejudices according to any dictionary definition.
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As we speak of poetical beauty, so ought we to speak of mathematical beauty and medical beauty. But we do not do so; and that reason is that we know well what is the object of mathematics, and that it consists in proofs, and what is the object of medicine, and that it consists in healing. But we do not know in what grace consists, which is the object of poetry.
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