How did Gerhard Gentzen’s sequent calculus contribute to proof theory?

Gerhard Gentzen made foundational contributions to proof theory through the introduction of the sequent calculus, a system designed to study the formal properties of logical inference.1

His work significantly advanced the field in the following ways:

• The Hauptsatz: Gentzen’s first Hauptsatz (fundamental theorem) demonstrated that proofs could be structured such that every step consists of subformulas of the final conclusion. This result established a method for transforming proofs into "normal forms," which remains a cornerstone of proof-theoretic research.1
• Consistency Proofs: Building on the project of David Hilbert to verify the consistency of mathematical systems, Gentzen successfully proved the consistency of arithmetic in 1936 by employing nonfinitistic assumptions.1
• Extraction of Information: By utilizing the normal forms of proofs derived from his calculus, logicians can extract deeper insights from logical consequences. A notable example is Craig’s interpolation theorem, which allows for the identification of an interpolation formula that explains why one proposition follows from another.1

Gentzen’s innovations provided the essential framework for analyzing the syntactic structure of proofs, influencing subsequent developments in both logic and computer science.1