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A Formally Verified Abstract Account of Gödel’s Incompleteness Theorems

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Automated Deduction – CADE 27 (CADE 2019)

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Abstract

We present an abstract development of Gödel’s incompleteness theorems, performed with the help of the Isabelle/HOL theorem prover. We analyze sufficient conditions for the theorems’ applicability to a partially specified logic. In addition to the usual benefits of generality, our abstract perspective enables a comparison between alternative approaches from the literature. These include Rosser’s variation of the first theorem, Jeroslow’s variation of the second theorem, and the Świerczkowski–Paulson semantics-based approach. As part of our framework’s validation, we upgrade Paulson’s Isabelle proof to produce a mechanization of the second theorem that does not assume soundness in the standard model, and in fact does not rely on any notion of model or semantic interpretation.

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Acknowledgments

We thank Bernd Buldt for his patient explanations on material in his monograph, and the reviewers for insightful comments and suggestions.

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Authors and Affiliations

  1. Department of Computer Science, Middlesex University, London, UK

    Andrei Popescu

  2. Institute of Information Security, Department of Computer Science, ETH Zürich, Zurich, Switzerland

    Dmitriy Traytel

Authors
  1. Andrei Popescu
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  2. Dmitriy Traytel
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Correspondence to Andrei Popescu or Dmitriy Traytel .

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Editors and Affiliations

  1. University of Lorraine, Villers-lès-Nancy, France

    Pascal Fontaine

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Popescu, A., Traytel, D. (2019). A Formally Verified Abstract Account of Gödel’s Incompleteness Theorems. In: Fontaine, P. (eds) Automated Deduction – CADE 27. CADE 2019. Lecture Notes in Computer Science(), vol 11716. Springer, Cham. https://doi.org/10.1007/978-3-030-29436-6_26

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  • DOI: https://doi.org/10.1007/978-3-030-29436-6_26

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