How does proof theory study formal logical systems?

Proof theory studies formal logical systems by focusing on their purely syntactical aspects, specifically the structure of rules of inference and the nature of the proofs conducted within those systems.1

Key methods and objectives include:

• Formalization: Systems are constructed using a finite or countable set of axioms and formal rules, aiming to derive all truths of a specific logical domain as theorems.1
• Consistency Analysis: Rooted in the project of David Hilbert, proof theory attempts to demonstrate the consistency of mathematical theories by showing that no contradiction can be derived from their formal axioms.1
• Structural Analysis: Logicians like Gerhard Gentzen developed systems such as the sequent calculus to study the properties of proofs. Gentzen’s fundamental theorems demonstrated that proofs can be performed in "normal forms," where steps are continuous parts of later ones.1
• Information Extraction: Beyond verifying consistency, proof theory allows for the extraction of significant information, such as Craig’s interpolation theorem, which explains why one proposition follows from another.1