Proof Theory is a branch of mathematical logic that focuses on the structure and nature of mathematical proofs. It studies how statements can be proven true or false using formal systems, which consist of axioms and rules of inference. By analyzing these proofs, mathematicians can understand the foundations of mathematics and the relationships between different mathematical statements. One of the key goals of proof theory is to establish the consistency and completeness of various logical systems. This involves showing that if a statement can be proven within a system, it is true in all interpretations of that system. Gödel's Incompleteness Theorems are significant results in this area, demonstrating limitations in formal systems.