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.
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Notes
- 1.
The authors are most grateful to the anonymous referee for hinting at this issue.
- 2.
This and other choices of terminology typical for constructive settings are made to prepare for Sect. 3.3.
- 3.
We could equally have worked for and with propositional logic, with arbitrary formulas in place of sentences.
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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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Ciraulo, F., Rinaldi, D., Schuster, P. (2016). Lindenbaum’s Lemma via Open Induction. In: Kahle, R., Strahm, T., Studer, T. (eds) Advances in Proof Theory. Progress in Computer Science and Applied Logic, vol 28. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-29198-7_3
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