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Incompleteness, Undecidability and Automated Proofs

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Computer Algebra in Scientific Computing (CASC 2016)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 9890))

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Abstract

Incompleteness and undecidability have been used for many years as arguments against automatising the practice of mathematics. The advent of powerful computers and proof-assistants – programs that assist the development of formal proofs by human-machine collaboration – has revived the interest in formal proofs and diminished considerably the value of these arguments.

In this paper we discuss some challenges proof-assistants face in handling undecidable problems – the very results cited above – using for illustrations the generic proof-assistant Isabelle.

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Notes

  1. 1.

    Included in the short radio presentation, see [24].

  2. 2.

    The words are engraved on Hilbert’s tombstone in Göttingen. This is a triple irony: their use as an epitaph, the fact that the day before the talk, Hilbert’s optimism was undermined by Gödel’s presentation of the incompleteness theorem, whose exceptional significance was, with the exception of John von Neumann, completely missed by the audience.

  3. 3.

    The approximate equivalent of all the digitised texts held by the US Library of Congress.

  4. 4.

    Historically, the syntactic class of partial functions constructed recursively is called partially recursive functions, see [25, 29]. This class coincides with the semantic partial functions implementable by standard models of computation (Turing machines, URMs, the \(\lambda \)-calculus etc.) – the partially computable functions.

  5. 5.

    In Isabelle, the @ symbol indicates concatenation of lists. Also note that this definition of halts assumes functions with some number of inputs and a single output.

  6. 6.

    Of course it should be noted that if an input is not within that domain of a function, Isabelle’s attempt to evaluate is likely not to terminate. However, consequential strange behaviours can be observed, such as in \(g_2\).

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Author information

Authors and Affiliations

  1. Department of Computer Science, University of Auckland, Auckland, New Zealand

    Cristian S. Calude & Declan Thompson

  2. Department of Philosophy, Stanford University, Stanford, USA

    Declan Thompson

Authors
  1. Cristian S. Calude
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  2. Declan Thompson
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Correspondence to Cristian S. Calude .

Editor information

Editors and Affiliations

  1. Laboratory of Information Technologies, Joint Institute of Nuclear Research, Dubna, Russia

    Vladimir P. Gerdt

  2. Institut für Mathematik, Universität Kassel, Kassel, Germany

    Wolfram Koepf

  3. Institut für Mathematik, Universität Kassel, Kassel, Germany

    Werner M. Seiler

  4. Institute of Theoretical and Applied Mechanics, Russian Academy of Sciences, Novosibirsk, Russia

    Evgenii V. Vorozhtsov

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Calude, C.S., Thompson, D. (2016). Incompleteness, Undecidability and Automated Proofs. In: Gerdt, V., Koepf, W., Seiler, W., Vorozhtsov, E. (eds) Computer Algebra in Scientific Computing. CASC 2016. Lecture Notes in Computer Science(), vol 9890. Springer, Cham. https://doi.org/10.1007/978-3-319-45641-6_10

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  • DOI: https://doi.org/10.1007/978-3-319-45641-6_10

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