Abstract
We analyze Ekeland’s variational principle in the context of reverse mathematics. We find that that the full variational principle is equivalent to Pi ^1_1text{- }mathsf {CA}_0, a strong theory of second-order arithmetic, while natural restrictions (e.g. to compact spaces or to continuous functions) yield statements equivalent to weak König’s lemma (mathsf {WKL}_0) and to arithmetical comprehension (mathsf {ACA}_0). We also find that the localized version of Ekeland’s variational principle is equivalent to Pi ^1_1text{- }mathsf {CA}_0, even when restricted to continuous functions. This is a rare example of a statement about continuous functions having great logical strength.
Highlights
There are many advantages to determining these minimal assumptions, among them aiding in extracting computational content from theorems whose original proofs were non-constructive. It is desirable from a methodological perspective to avoid strong set-theoretic assumptions when possible, and advances in reverse mathematics have shown us that a large portion of known mathematics can be carried out within a relatively small fragment of second-order arithmetic [14]
Second-order arithmetic extends the language of Peano arithmetic by adding variables and quantifiers ranging over sets of natural numbers; this is sufficient to formalize many familiar concepts from analysis, including real numbers, continuous functions, and open and closed sets
WKL0, ACA0, and 11-CA0, including what is, to the best of our knowledge, the first statement about continuous functions stemming from analysis that is equivalent to Before diving into formal systems, let us discuss the variational principle in more detail and sketch a proof
Summary
The field of reverse mathematics, introduced by Friedman [10], aims to identify the minimal foundational assumptions required to prove specific results from several mathematical fields, including mathematical analysis. Our goal in this article is to study Ekeland’s variational principle [5] in the context of reverse mathematics This principle states that under certain conditions, lower semicontinuous functions on complete metric spaces always attain ‘approximate minima,’. A posteriori the analysis of Ekeland’s variational principle is even more interesting and quite surprising: as we will see, natural restrictions of the result (e.g. to compact spaces, to continuous f ) yield statements equivalent over RCA0 to each of WKL0, ACA0, and 11-CA0, including what is, to the best of our knowledge, the first statement about continuous functions stemming from analysis that is equivalent to Before diving into formal systems, let us discuss the variational principle in more detail and sketch a proof. Ekeland’s variational principle states that f has points that are in a sense approximate local minima, and we
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