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

• David Fernández-Duque ^[1] ; Paul Shafer ^[2] ; Keita Yokoyama ^[3]
  1. [1] Ghent University

     Ghent University

     Arrondissement Gent, Bélgica

     
  2. [2] University of Leeds

     University of Leeds

     Reino Unido

     
  3. [3] Japan Advanced Institute of Science and Technology

     Japan Advanced Institute of Science and Technology

     Japón

     

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• Localización: Selecta Mathematica, New Series, ISSN 1022-1824, Vol. 26, Nº. 5, 2020
• Idioma: inglés
• DOI: 10.1007/s00029-020-00597-z
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• Resumen

  • 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\text{- }\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_1\text{- }\mathsf {CA}_0, even when restricted to continuous functions. This is a rare example of a statement about continuous functions having great logical strength.