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arXiv · 1902.03915

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

David Fernández-Duque·Paul Shafer·Keita Yokoyama
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Abstract

We analyze Ekeland's variational principle in the context of reverse mathematics. We find that that the full variational principle is equivalent to $\Pi^1_1$-${\sf CA}_0$, a strong theory of second-order arithmetic, while natural restrictions (e.g.~to compact spaces or continuous functions) yield statements equivalent to weak K\"onig's lemma (${\sf WKL}_0$) and to arithmetical comprehension (${\sf ACA}_0$). We also find that the localized version of Ekeland's variational principle is equivalent to $\Pi^1_1$-${\sf CA}_0$ 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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David Fernández-Duque, Paul Shafer, Keita Yokoyama. 2019-02-11. Ekeland's variational principle in weak and strong systems of arithmetic. https://arxiv.org/abs/1902.03915

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