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arXiv · 2512.12892

Non-Compact Proofs

Sergei Artemov
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Abstract

Non-compact proofs are a class of reasoning that is used in mathematics but overlooked in the analysis of (un)provability of consistency. We focus on proofs of arithmetical statements (*) "for any natural number n, F(n)." A proof of (*) is called compact if all proofs of F(n)'s for n=0,1,2,... fit into some finitely axiomatized fragment of Peano Arithmetic PA. An example of non-compact reasoning is given by the standard proof of Mostowski's 1952 reflexivity theorem: PA proves the consistency of its finite fragments. It turns out that G\"odel's second incompleteness theorem prohibits compact proofs and does not rule out non-compact proofs of PA-consistency formalizable in PA. This explains how the recent proof of PA-consistency in PA works. It essentially formalizes in PA the explicit version of Mostowski's proof, which is not in the scope of G\"odel's theorem.

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Sergei Artemov. 2025-12-15. Non-Compact Proofs. https://arxiv.org/abs/2512.12892

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