# Proof Theory

[https://en.wikipedia.org/wiki/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](../logic-inference/rule-of-inference) of the logical system. As such, proof theory is syntactic in nature, in contrast to [model theory](proof-theory), 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.
