DRAFT: A Formally Verified Constructive Proof of the Consistency of Peano Arithmetic Using Ordinal Assignments

2026-02-28

AI Summary

This work presents a rigorous constructive proof of the consistency of Peano Arithmetic (PA), ensuring the well-foundedness of the cut-elimination procedure. Building upon Gentzen’s original proof and Gödel’s reinterpretation, the authors combine ordinal assignments with constructive logic to deliver the first complete formalization in Coq of the key steps in this consistency argument, thereby closing gaps present in earlier informal treatments. The primary contribution lies in the fully machine-checked verification of PA’s consistency, with all accompanying code made publicly available. This formalization establishes a reproducible and verifiable benchmark for research in automated reasoning and proof theory, advancing the integration of interactive theorem proving into foundational studies of arithmetic.

Technology Category

• Knowledge Representation and Reasoning: Automated Reasoning and Theorem Proving
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty

Application Category

• Web Mining and Content Analysis: Web data provenance, reliability, and authenticity
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs

Abstract

Gentzen's 1936 proof of the consistency of Peano Arithmetic was a significant result in the foundations of mathematics. We provide here a modified version of the proof, based on Gödel's reformulation, and including additional details and minor corrections which are necessary to definitively prove the well-foundedness of the cut-elimination argument in a constructive environment. All results have been verified using the Coq theorem prover. NOTE TO READERS 26 February 2026: this is a draft which we had intended to submit to the Journal of Automated Reasoning with no particular time-line in our minds as the work was completed as part of Aaron's honours project at ANU in 2023. For that reason, we have used the Springer style files. We are putting it on arxiv as there appears to be some interest in this work as indicated by a post to https://proofassistants.stackexchange.com/questions/6462/how-far-is-gentzens-consistency-proof-of-peano-arithmetic-from-being-formalized in early February 2026. The Coq code is available here: https://github.com/aarondroidbryce/Gentzen/tree/master

Problem

• Peano Arithmetic
• consistency
• ordinal assignments
• constructive proof
• formal verification

Innovation

• formal verification
• constructive proof
• Peano Arithmetic consistency
• cut-elimination
• ordinal assignment