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arXiv · 2607.15958

Logical Metatheorems for Abstract Spaces axiomatized in Positive Bounded Logic II: Metric spaces and the model-theoretic uniformity principle

Ulrich Kohlenbach·Morenikeji Neri·Jin Wei
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Abstract

We extend the proof-theoretic treatment of uniform bound extraction from normed structures axiomatized in positive bounded logic [Advances in Mathematics, 290:503-551, 2016] (as developed for the model theory of Banach spaces) to the more general setting of abstract metric structures, including discrete structures viewed as classical first-order models. In particular, we establish uniform bound extraction theorems for our generalized framework for $\forall\exists$-sentences whose matrix is the negation of (an embedding of) a formula in positive bounded logic, whose proofs use saturation. In this way, we provide a formal explanation for the successes in the extraction of uniform bounds from nonstandard proofs given in [Advances in Mathematics, 343:567-623, 2019], which had informally followed the perspective of the monotone functional interpretation. As an application of the formal framework we develop, we provide novel explicit bounds for a structural theorem for stable subsets of groups given in [Mathematical Proceedings of the Cambridge Philosophical Society, 168(2):405-413, 2020].

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Ulrich Kohlenbach, Morenikeji Neri, Jin Wei. 2026-07-17. Logical Metatheorems for Abstract Spaces axiomatized in Positive Bounded Logic II: Metric spaces and the model-theoretic uniformity principle. https://arxiv.org/abs/2607.15958

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