Ekeland's variational principle in weak and strong systems of arithmetic

2019/02/11 by David Fernández-Duque, David Fernández–Duque, Paul Shafer +4
Computer Science · Mathematics · #Advanced Topology and Set Theory #Computability, Logic, AI Algorithms #FOS: Mathematics #Logic (math.LO) #Mathematical and Theoretical Analysis #math.LO

paper · pdf · doi:10.48550/arxiv.1902.03915

openalex publication_date 2019/02/11 · arxiv created 2020/09/15 · arxiv updated 2020/09/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We analyze Ekeland's variational principle in the context of reverse mathematics. We find that that the full variational principle is equivalent to Π¹₁-\sf CA₀, a strong theory of second-order arithmetic, while natural restrictions (e.g.~to compact spaces or continuous functions) yield statements equivalent to weak König's lemma (\sf WKL₀) and to arithmetical comprehension (\sf ACA₀). We also find that the localized version of Ekeland's variational principle is equivalent to Π¹₁-\sf CA₀ even when restricting to continuous functions. This is a rare example of a statement about continuous functions having great logical strength.

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