# Proof Theory

**Type:** Concept  
**Domain:** foundations-logic  
**Codex URL:** /mathematics/foundations-logic/proof-theory/  
**Entry status:** Live — v1.0 (2026-07-09)

## Summary
Studies mathematical proofs as formal objects. Founded by Hilbert 1920s; transformed by Gödel 1931. Key tools: cut elimination (Gentzen 1934), ordinal analysis, reverse mathematics, proof complexity.

## Sources
- [Tier 1] Gentzen, G. (1935). Mathematische Zeitschrift, 39, 176-210.
- [Tier 1] Simpson, S.G. (2009). Subsystems of Second Order Arithmetic. 2nd ed.
- [Tier 2] Buss, S.R. (ed.) (1998). Handbook of Proof Theory.
- [Tier 3] Negri, S. and von Plato, J. (2001). Structural Proof Theory.

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*Mathematics Codex entry v1.0 — added 2026-07-09 — thecodex.expert/mathematics/*