![]() ![]() Theories of logic were developed in many cultures in history, including China, India, Greece and the Islamic world. The first half of the 20th century saw an explosion of fundamental results, accompanied by vigorous debate over the foundations of mathematics. "Mathematical logic, also called 'logistic', 'symbolic logic', the ' algebra of logic', and, more recently, simply 'formal logic', is the set of logical theories elaborated in the course of the last Nineteenth Century with the aid of an artificial notation and a rigorously deductive method." Before this emergence, logic was studied with rhetoric, with calculationes, through the syllogism, and with philosophy. Mathematical logic emerged in the mid-19th century as a subfield of mathematics, reflecting the confluence of two traditions: formal philosophical logic and mathematics. These foundations use toposes, which resemble generalized models of set theory that may employ classical or nonclassical logic. Because of its applicability in diverse fields of mathematics, mathematicians including Saunders Mac Lane have proposed category theory as a foundational system for mathematics, independent of set theory. The mathematical field of category theory uses many formal axiomatic methods, and includes the study of categorical logic, but category theory is not ordinarily considered a subfield of mathematical logic. The method of forcing is employed in set theory, model theory, and recursion theory, as well as in the study of intuitionistic mathematics. Gödel's incompleteness theorem marks not only a milestone in recursion theory and proof theory, but has also led to Löb's theorem in modal logic. The borderlines amongst these fields, and the lines separating mathematical logic and other fields of mathematics, are not always sharp. Each area has a distinct focus, although many techniques and results are shared among multiple areas. proof theory and constructive mathematics (considered as parts of a single area).Īdditionally, sometimes the field of computational complexity theory is also included as part of mathematical logic.The Handbook of Mathematical Logic in 1977 makes a rough division of contemporary mathematical logic into four areas: 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. 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. Results of Kurt Gödel, Gerhard Gentzen, and others provided partial resolution to the program, and clarified the issues involved in proving consistency. In the early 20th century it was shaped by David Hilbert's program to prove the consistency of foundational theories. This study began in the late 19th century with the development of axiomatic frameworks for geometry, arithmetic, and analysis. Since its inception, mathematical logic has both contributed to and been motivated by the study of foundations of mathematics. However, it can also include uses of logic to characterize correct mathematical reasoning or to establish foundations of mathematics. Research in mathematical logic commonly addresses the mathematical properties of formal systems of logic such as their expressive or deductive power. Major subareas include model theory, proof theory, set theory, and recursion theory. Mathematical logic is the study of formal logic within mathematics. ![]()
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