License: Creative Commons Attribution 4.0 International license (CC BY 4.0)
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DOI: 10.4230/LIPIcs.CSL.2023.30
URN: urn:nbn:de:0030-drops-174911
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Kirst, Dominik ; Peters, Benjamin

Gödel’s Theorem Without Tears - Essential Incompleteness in Synthetic Computability

LIPIcs-CSL-2023-30.pdf (0.7 MB)


Gödel published his groundbreaking first incompleteness theorem in 1931, stating that a large class of formal logics admits independent sentences which are neither provable nor refutable. This result, in conjunction with his second incompleteness theorem, established the impossibility of concluding Hilbert’s program, which pursued a possible path towards a single formal system unifying all of mathematics. Using a technical trick to refine Gödel’s original proof, the incompleteness result was strengthened further by Rosser in 1936 regarding the conditions imposed on the formal systems.
Computability theory, which also originated in the 1930s, was quickly applied to formal logics by Turing, Kleene, and others to yield incompleteness results similar in strength to Gödel’s original theorem, but weaker than Rosser’s refinement. Only much later, Kleene found an improved but far less well-known proof based on computational notions, yielding a result as strong as Rosser’s.
In this expository paper, we work in constructive type theory to reformulate Kleene’s incompleteness results abstractly in the setting of synthetic computability theory and assuming a form of Church’s thesis, an axiom internalising the fact that all functions definable in such a setting are computable. Our novel, greatly condensed reformulation showcases the simplicity of the computational argument while staying formally entirely precise, a combination hard to achieve in typical textbook presentations. As an application, we instantiate the abstract result to first-order logic in order to derive essential incompleteness and, along the way, essential undecidability of Robinson arithmetic.
This paper is accompanied by a Coq mechanisation covering all our results and based on existing libraries of undecidability proofs and first-order logic, complementing the extensive work on mechanised incompleteness using the Gödel-Rosser approach. In contrast to the related mechanisations, our development follows Kleene’s ideas and utilises Church’s thesis for additional simplicity.

BibTeX - Entry

  author =	{Kirst, Dominik and Peters, Benjamin},
  title =	{{G\"{o}del’s Theorem Without Tears - Essential Incompleteness in Synthetic Computability}},
  booktitle =	{31st EACSL Annual Conference on Computer Science Logic (CSL 2023)},
  pages =	{30:1--30:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-264-8},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{252},
  editor =	{Klin, Bartek and Pimentel, Elaine},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{},
  URN =		{urn:nbn:de:0030-drops-174911},
  doi =		{10.4230/LIPIcs.CSL.2023.30},
  annote =	{Keywords: incompleteness, undecidability, synthetic computability theory}

Keywords: incompleteness, undecidability, synthetic computability theory
Collection: 31st EACSL Annual Conference on Computer Science Logic (CSL 2023)
Issue Date: 2023
Date of publication: 01.02.2023
Supplementary Material: To seamlessly integrate the mechanisation with the written text, each formal statement in the PDF version of this paper is hyperlinked with HTML documentation of the Coq files (signalled via a small Coq symbol).
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Software (Source Code):

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