Proof theory

Proof theory studies formal proofs as mathematical objects, examining their structure, transformations, and the rules by which conclusions are derived from assumptions.

Formal system: A specified language together with rules for deriving or interpreting its expressions. Proof theory studies derivations inside systems defined by explicit languages and rules.
Cut-elimination theorem: A theorem stating that certain proof systems admit derivations without applications of the cut rule. It is a central structural result about transforming proofs into simpler forms.
Hilbert's program: David Hilbert's project to justify mathematics through formalization and finitary consistency proofs. Its goals prompted major proof-theoretic work on consistency and formal systems.
Consistency proof: A mathematical argument that a formal system does not derive a contradiction. Constructing and calibrating consistency proofs is a longstanding proof-theoretic aim.
Model theory: The study of mathematical structures that satisfy formal languages and theories. It investigates interpretation and truth in structures, complementing proof-centered analysis.
Inference rule: A rule that licenses deriving a conclusion from specified premises. Inference rules determine which steps count as valid in a formal derivation.
Normalization (proof theory): The process of transforming a proof into a canonical form by eliminating detours. Normalization explains canonical proof forms in systems such as natural deduction.
David Hilbert: A German mathematician whose work shaped geometry, algebra, and the foundations of mathematics. His formalist program helped set proof theory's early research agenda.
Curry–Howard correspondence: A correspondence linking propositions with types and proofs with programs. It reveals computational content in formal derivations.
Computability theory: The study of which problems can be solved by algorithms and what resources computation requires. It focuses on effective procedures, though proof systems can encode computational limits.

Linked from 25 pages