Editing Proof theory (section)

{{short description|Branch of mathematical logic}}
'''Proof theory''' is a major branch<ref name=wang>According to {{harvtxt|Wang|1981|pp=3–4}}, proof theory is one of four domains mathematical logic, together with [[model theory]], [[axiomatic set theory]], and [[recursion theory]]. {{harvtxt|Barwise|1977}} consists of four corresponding parts, with part D being about "Proof Theory and Constructive Mathematics".</ref> of [[mathematical logic]] and [[theoretical computer science]] within which [[Mathematical proof|proof]]s are treated as formal [[mathematical object]]s, facilitating their analysis by mathematical techniques. Proofs are typically presented as [[Recursive data type|inductively defined]] [[data structures]] such as [[list (computer science)|lists]], boxed lists, or [[Tree (data structure)|tree]]s, which are constructed according to the [[axiom]]s and [[rule of inference|rules of inference]] of a given logical system. Consequently, proof theory is [[syntax (logic)|syntactic]] in nature, in contrast to [[model theory]], which is [[Formal semantics (logic)|semantic]] in nature.

Some of the major areas of proof theory include [[structural proof theory]], [[ordinal analysis]], [[provability logic]], [[reverse mathematics]], [[proof mining]], [[automated theorem proving]], and [[proof complexity]]. Much research also focuses on applications in computer science, linguistics, and philosophy.