License: Creative Commons Attribution 3.0 Unported license (CC BY 3.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.CSL.2021.21
URN: urn:nbn:de:0030-drops-134552
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2021/13455/
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Forster, Yannick

Church’s Thesis and Related Axioms in Coq’s Type Theory

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LIPIcs-CSL-2021-21.pdf (0.6 MB)

Abstract

"Church’s thesis" (CT) as an axiom in constructive logic states that every total function of type ℕ → ℕ is computable, i.e. definable in a model of computation. CT is inconsistent both in classical mathematics and in Brouwer’s intuitionism since it contradicts weak Kőnig’s lemma and the fan theorem, respectively. Recently, CT was proved consistent for (univalent) constructive type theory.
Since neither weak Kőnig’s lemma nor the fan theorem is a consequence of just logical axioms or just choice-like axioms assumed in constructive logic, it seems likely that CT is inconsistent only with a combination of classical logic and choice axioms. We study consequences of CT and its relation to several classes of axioms in Coq’s type theory, a constructive type theory with a universe of propositions which proves neither classical logical axioms nor strong choice axioms.
We thereby provide a partial answer to the question as to which axioms may preserve computational intuitions inherent to type theory, and which certainly do not. The paper can also be read as a broad survey of axioms in type theory, with all results mechanised in the Coq proof assistant.

BibTeX - Entry

@InProceedings{forster:LIPIcs:2021:13455,
  author =	{Yannick Forster},
  title =	{{Church’s Thesis and Related Axioms in Coq’s Type Theory}},
  booktitle =	{29th EACSL Annual Conference on Computer Science Logic (CSL 2021)},
  pages =	{21:1--21:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-175-7},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{183},
  editor =	{Christel Baier and Jean Goubault-Larrecq},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/opus/volltexte/2021/13455},
  URN =		{urn:nbn:de:0030-drops-134552},
  doi =		{10.4230/LIPIcs.CSL.2021.21},
  annote =	{Keywords: Church’s thesis, constructive type theory, constructive reverse mathematics, synthetic computability theory, Coq}
}

Keywords: Church’s thesis, constructive type theory, constructive reverse mathematics, synthetic computability theory, Coq
Collection: 29th EACSL Annual Conference on Computer Science Logic (CSL 2021)
Issue Date: 2021
Date of publication: 13.01.2021
Supplementary Material: https://github.com/uds-psl/churchs-thesis-coq

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