Proof theory is a branch of mathematical logic that represents proofs as formal mathematical objects, facilitating their analysis by mathematical techniques. Proofs are typically presented as inductively-defined data structures such as plain lists, boxed lists, or trees, which are constructed according to the axioms and rules of inference of the logical system. As such, proof theory is syntactic in nature, in contrast to model theory, which is semantic in nature. Together with model theory, axiomatic set theory, and recursion theory, proof theory is one of the so-called four pillars of the foundations of mathematics.

Proof theory is important in philosophical logic, where the primary interest is in the idea of a proof-theoretic semantics, an idea which depends upon technical ideas in structural proof theory to be feasible.

• People (5)
• Provability and Interpretability Logics@ (9)

See also:

• Science: Math: Logic and Foundations: Computational Logic: Automated Reasoning (1)

• Proof Theory
  open-site.org
  Open Encyclopedia entry. Hierarchically organized by subtopics.
• Proof Theory as an Alternative to Model Theory
  lix.polytechnique.fr
  Short article by Dale Miller, arguing that logic programming languages should base their semantics on proof theory, not model theory.
• The Epsilon Calculus
  plato.stanford.edu
  Discussion of David Hilbert's development of this type of logical formalism with emphasis on proof-theoretic methods.