Mathematical logic
lastmod 2014-11-02created 2014-11-021 edits
Mathematical logic is a subfield of mathematics and logic, closely related to computer science and philosophical logic. The field includes the mathematical study of logic and of the application of formal logic to other fields, and, taken as a whole, the study of the expressive power of formal systems and the deductive power of formal proof systems.
Mathematical logic is often divided into subfields such as set theory, model theory, recursion theory, proof theory, and constructive mathematics. These fields share a common basis in fundamental logical results such as first-order logic and definability.
Since its emergence, mathematical logic has continually influenced and been influenced by the study of the foundations of mathematics. This study began in the late 19th century with the development of axiomatic frameworks for geometry, algebra, and analysis. In the early 20th century it was shaped by David Hilbert's program to prove the consistency of foundational theories. Kurt Gödel, Gerhard Gentzen, and others offered partial solutions to that program and clarified the issues involved in proving consistency. Although some theorems cannot be proved within the axiom systems of set theory, work in set theory showed that almost all ordinary mathematics can be formalized in terms of sets. Recent work in the foundations of mathematics often focuses on finding which parts of mathematics can be formalized in particular formal systems, rather than on finding a theory in which all of mathematics can be developed.