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Direct Proofs of Lindenbaum Conditionals

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Abstract

We discuss the problem raised by Miller (Log Univers 1:183–199, 2007) to re-prove the well-known equivalences of some Lindenbaum theorems for deductive systems (each equivalent to the Axiom of Choice) without an application of the Axiom of Choice. We present five special constructions of deductive systems, each of them providing some partial solutions to the mathematical problem. We conclude with a short discussion of the underlying philosophical problem of deciding, whether a given proof satisfies our demand that the Axiom of Choice is not applied.

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Authors and Affiliations

  1. Department of Computer Science, University of Tübingen, Sand 13, 72076, Tübingen, Germany

    René Gazzari

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  1. René Gazzari
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Correspondence to René Gazzari.

Additional information

This work was supported by the ESF EUROCORES project “Dialogical Foundations of Semantics” and by the French-German ANR-DFG Project “Hypothetical Reasoning” (DFG grants Schr275/15 and Schr275/16, FCT grant SFRH/BI/33955/2009).

I am grateful to my supervisors Peter Hauck, Reinhard Kahle and Peter Schroeder-Heister as well as to several anonymous reviewers. Their detailed comments and suggestions helped very much to improve this paper. Special thanks are due to David Miller. He motivated me to follow my line of research and supported my work with great effort.

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Gazzari, R. Direct Proofs of Lindenbaum Conditionals. Log. Univers. 8, 321–343 (2014). https://doi.org/10.1007/s11787-013-0081-1

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  • DOI: https://doi.org/10.1007/s11787-013-0081-1

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