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Lindenbaum’s Lemma via Open Induction

  • Francesco CirauloEmail author
  • Davide Rinaldi
  • Peter Schuster
Chapter
Part of the Progress in Computer Science and Applied Logic book series (PCS, volume 28)

Abstract

With Raoult’s Open Induction in place of Zorn’s Lemma, we do a perhaps more perspicuous proof of Lindenbaum’s Lemma for not necessarily countable languages of first-order predicate logic. We generally work for and with classical logic, but say what can be achieved for intuitionistic logic, which prompts the natural generalizations for distributive and complete lattices.

Notes

Acknowledgments

The research that has led to this note was carried out within the project “Abstract Mathematics for Actual Computation: Hilbert’s Program in the 21st Century” funded by the John Templeton Foundation, and within two of the European Union’s Marie Curie projects: the Initial Training Network “MALOA: From Mathematical Logic to Applications” and the International Research Exchange Scheme project “CORCON: Correctness by Construction”. The final version of the present note was prepared when the third author was visiting the Munich Center for Mathematical Philosophy: upon kind invitation by Hannes Leitgeb and with a research fellowship “Erneuter Aufenthalt” by the Alexander-von-Humboldt Foundation. All authors wish to thank Thierry Coquand, Volker Halbach, Kentaro Fujimoto, Giovanni Sambin and the anonymous referee for useful hints and constructive critique. Last but not least, the third author would like to express his gratitude to Gerhard Jäger for now more than a decade of encouragement, support and hospitality.

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Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  • Francesco Ciraulo
    • 1
    Email author
  • Davide Rinaldi
    • 2
  • Peter Schuster
    • 3
  1. 1.Dipartimento di MatematicaUniversità degli Studi di PadovaPadovaItaly
  2. 2.Department of Pure MathematicsUniversity of LeedsLeeds LS2 9JTEngland
  3. 3.Dipartimento di InformaticaUniversità degli Studi di VeronaVeronaItaly

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