A subfield of mathematical logic known as proof theory provides proofs as formal mathematical objects to make their investigation by mathematical methods easier. Proofs are often provided as inductively defined data structures that are built in accordance with the axioms and rules of inference of the logical system, such as simple lists, boxed lists, or trees. Therefore, unlike model theory, which is semantic in character, the proof theory is syntactic in nature. Proof theory is one of the so-called four pillars of mathematics, along with model, axiomatic set, and recursion theories. Structural proof theory, ordinal analysis, provability logic, reverse mathematics, proof mining, automated theorem proving, and proof complexity are some of the main subfields of proof theory. Applications in computer science, linguistics, and philosophy are also the subject of a lot of research. This book is intended to provide a basic review of the subject and to give you the organized knowledge you need to become familiar with it for the least amount of money possible. The conversation is at Wikipedia’s level. Since the edited articles contain the contributions of numerous competent people and some of the most current general information on the topic as of the publication date, the accuracy and knowledge are of an international perspective.