prooftheoretic

Prooftheoretic is an adjective used in logic and philosophy to describe things pertaining to proof theory, the study of mathematical proofs and their formal properties. It encompasses the analysis of what counts as a proof, the structure of proofs, and the notion of provability in formal systems.

Historically, proof theory emerged from the early 20th century program of Hilbert to establish the consistency of mathematics by finitistic means. A turning point was Gerhard Gentzen's work in the 1930s on sequent calculus and natural deduction, yielding the cut-elimination theorem and a proof of the consistency of Peano arithmetic using transfinite induction up to epsilon_0. These ideas laid the foundation for modern prooftheoretic methods and ordinal analysis.

Core topics include formal systems described by inference rules, structural rules, and the syntactic analysis of proofs. Sequent calculus, natural deduction, and structural proof theory study how proofs can be transformed, simplified, and normalized. Proof-theoretic semantics, a related program, seeks to define the meaning of logical connectives by the rules governing proofs rather than by truth conditions.

In contemporary work, prooftheoretic methods inform the design of logical frameworks, type theories, and constructive mathematics, as well as formal verification and automated reasoning. The field intersects with computer science through the Curry-Howard correspondence, which connects proofs with programs and types.

Prominent figures include Gerhard Gentzen, Dag Prawitz, and Takeuti, with ongoing contributions from researchers in structural proof theory and ordinal analysis.