Independence Day I gonna need another beer to wash down this beer Flag Sometimes, a state wishing to achieve independence from a dominating power will issue a declaration of independence; the earliest surviving example is Scotland’s Declaration of Arbroath in 1320, with the most recent example being Azawad’s declaration of independence in 2012. Declaring independence and attaining it, however, are quite different. A well-known successful example is the U.S. Declaration of Independence issued in 1776. The dates of established independence (or, less commonly, the commencement of revolution), are typically celebrated as a national holiday known as an independence Day.
Independence Day I gonna need another beer to wash down this beer Flag 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.
Independence Day I gonna need another beer to wash down this beer Flag Each area has a distinct focus, although many techniques and results are shared among multiple areas. The borderlines amongst these fields, and the lines separating mathematical logic and other fields of mathematics, are not always sharp. 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 method of forcing is employed in set theory, model theory, and recursion theory, as well as in the study of intuitionistic mathematics.
Independence Day I gonna need another beer to wash down this beer Flag 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. 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. These foundations use toposes, which resemble generalized models of set theory that may employ classical or nonclassical logic.
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