2019/05/23 by Tadeusz Litak, Litak, Tadeusz, Albert Visser +1 · 1 citation
Computer Science · #Advanced Algebra and Logic #FOS: Computer and information sciences #Logic in Computer Science (cs.LO) #Logic, Reasoning, and Knowledge #Logic, programming, and type systems
paper · pdf · doi:10.48550/arxiv.1905.09450
openalex publication_date 2019/05/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Our paper is the first study of what one might call "reverse mathematics of explicit fixpoints". We study two methods of constructing such fixpoints for formulas whose principal connective is the intuitionistic Lewis arrow. Our main motivation comes from metatheory of constructive arithmetic, but the systems in question allows several natural semantics. The first of these methods, inspired by de Jongh and Visser, turns out to yield a well-understood modal system. The second one by de Jongh and Sambin, seemingly simpler, leads to a modal theory that proves harder to axiomatize in an elegant way. Apart from showing that both theories are incomparable, we axiomatize their join and investigate several subtheories, whose axioms are obtained as fixpoints of simple formulas. We also show that they are extension stable, that is, their validity in the corresponding preservativity logic of a given arithmetical theory transfer to its finite extensions.