> For the complete documentation index, see [llms.txt](https://mandober.gitbook.io/math-debrief/llms.txt). Markdown versions of documentation pages are available by appending `.md` to page URLs; this page is available as [Markdown](https://mandober.gitbook.io/math-debrief/300-logic/380-proof-theory/proof-theory.md).

# Proof Theory

<https://en.wikipedia.org/wiki/Proof_theory>

Proof theory is a major branch of mathematical logic that represents proofs as formal mathematical objects, facilitating their analysis by mathematical techniques.

Proofs are typically presented as inductively-defined data structures such as lists or trees, which are constructed according to the axioms and [rules of inference](/math-debrief/300-logic/logic-inference/rule-of-inference.md) of the logical system. As such, proof theory is syntactic in nature, in contrast to [model theory](/math-debrief/300-logic/380-proof-theory/proof-theory.md), which is 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
* proof complexity

Much research also focuses on applications in computer science, linguistics, and philosophy.