Mathematical logic is a subfield of
mathematics with close connections to
computer science and
philosophical logic.
[1] 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 subfields of set theory, model theory, recursion theory, and proof theory and constructive mathematics. These areas share basic results on logic, particularly first-order logic, and definability.
Since its inception, mathematical logic has 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, rather than trying to find theories in which all of mathematics can be developed.
Mathematical logic emerged in the mid-19th century as a subfield of mathematics independent of the traditional study of logic (Ferreirós 2001, p. 443). Before this emergence, logic was studied with rhetoric, through the syllogism, and with philosophy. The first half of the 20th century saw an explosion of fundamental results, accompanied by vigorous debate over the foundations of mathematics.
