Proof Theory in Mathematical Logic

Mathematical Logic is a powerful tool used by mathematicians and computer scientists alike to prove mathematical truths and build theories based on formalized languages. Proof Theory is one of the key areas within this field and deals with the notion of formally proving statements using rules and principles of reasoning.

At its core, proof theory involves constructing formulas from basic elements such as propositional variables, logical operators like 'and', 'or', 'implies' etc., and equality symbols. Rules govern how these formulas can interact to create new valid formulas which ultimately lead to proofs. These proofs are considered to be formal when they follow specific rules and conventions laid out in the system being studied. For example, you might need to show that (A \rightarrow B) follows from another statement (C), where '(A)' means some particular number is even while '(B)' signifies it is odd; consequently, if we assume C is true, A must also be true because there is a direct relation between them.

Proof theory has several important applications in mathematics and computer science. It allows us to express various properties and facts concerning algebraic structures, sets, functions, relations, topological spaces, groups, fields, algebras, rings, semigroups, abelian categories, ordered structures, vector spaces, linear mappings, partial orders, complete atomic Boolean algebras, Hilbert space representations of symmetric groups, Banach spaces, and more. This makes it especially useful when tackling complex problems such as those found across all branches of astronomy and astrophysics, physics, chemistry, biology, economics, political sciences, engineering, geography, psychology, sociology, anthropology, linguistics, philosophy, music theory, and natural history.

In summary, Proof Theory is a vital part of Mathematical Logic. Its main function lies in providing comprehensive methods through which to verify certain types of information. Additionally, its concepts have numerous real-world applications making it an indispensable branch of study within many scientific disciplines.