Editing Mathematical logic (section)

==Subfields and scope==
The ''Handbook of Mathematical Logic''{{sfnp|Barwise|1989}} in 1977 makes a rough division of contemporary mathematical logic into four areas:

#[[set theory]]
#[[model theory]]
#[[recursion theory]], and
#[[proof theory]] and [[constructive mathematics]] (considered as parts of a single area).

Additionally, sometimes the field of [[computational complexity theory]] is also included together with mathematical logic.<ref>{{Cite web |title=Logic and Computational Complexity {{!}} Department of Mathematics |url=https://math.ucsd.edu/research/logic-and-computational-complexity |access-date=2024-12-05 |website=math.ucsd.edu}}</ref><ref>{{Cite web|title=Computability Theory and Foundations of Mathematics / February, 17th – 20th, 2014 / Tokyo Institute of Technology, Tokyo, Japan|url=http://www.jaist.ac.jp/CTFM/CTFM2014/submissions/CTFM2014_booklet.pdf}}</ref> 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 (mathematics)|forcing]] is employed in set theory, model theory, and recursion theory, as well as in the study of intuitionistic mathematics.

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 [[topos]]es, which resemble generalized models of set theory that may employ classical or nonclassical logic.