The generalised continuum hypothesis implies the axiom of choice in Coq

Dominik Kirst; Felix Rech

doi:10.1145/3437992.3439932

The generalised continuum hypothesis implies the axiom of choice in Coq

Dominik Kirst, Felix Rech

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › peer-review

Abstract

We discuss and compare two Coq mechanisations of Sierpinski's result that the generalised continuum hypothesis (GCH) implies the axiom of choice (AC). The first version shows the result, originally stated in first-order ZF set-theory, for a higher-order set theory convenient to work with in Coq. The second version presents a corresponding theorem for Coq's type theory itself, concerning type-theoretic formulations of GCH and AC. Both versions rely on the classical law of excluded middle and extensionality assumptions but we localise the use of axioms where possible.

Original language	English
Title of host publication	CPP 2021 - Proceedings of the 10th ACM SIGPLAN International Conference on Certified Programs and Proofs, co-located with POPL 2021
Editors	Catalin Hritcu, Andrei Popescu
Publisher	Association for Computing Machinery, Inc
Pages	313-326
Number of pages	14
ISBN (Electronic)	9781450382991
DOIs	https://doi.org/10.1145/3437992.3439932
State	Published - 17 Jan 2021
Externally published	Yes
Event	10th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2021, co-located with POPL 2021 - Virtual, Online, Denmark Duration: 17 Jan 2021 → 19 Jan 2021

Publication series

Name	CPP 2021 - Proceedings of the 10th ACM SIGPLAN International Conference on Certified Programs and Proofs, co-located with POPL 2021

Conference

Conference	10th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2021, co-located with POPL 2021
Country/Territory	Denmark
City	Virtual, Online
Period	17/01/21 → 19/01/21

Keywords

Coq
axiom of choice
dependent type theory
generalised continuum hypothesis
higher-order set theory

ASJC Scopus subject areas

Artificial Intelligence
Computational Theory and Mathematics
Hardware and Architecture
Software
Information Systems and Management
Safety, Risk, Reliability and Quality
Logic
Education

Access to Document

10.1145/3437992.3439932

Cite this

Kirst, D., & Rech, F. (2021). The generalised continuum hypothesis implies the axiom of choice in Coq. In C. Hritcu, & A. Popescu (Eds.), CPP 2021 - Proceedings of the 10th ACM SIGPLAN International Conference on Certified Programs and Proofs, co-located with POPL 2021 (pp. 313-326). (CPP 2021 - Proceedings of the 10th ACM SIGPLAN International Conference on Certified Programs and Proofs, co-located with POPL 2021). Association for Computing Machinery, Inc. https://doi.org/10.1145/3437992.3439932

Kirst, Dominik ; Rech, Felix. / The generalised continuum hypothesis implies the axiom of choice in Coq. CPP 2021 - Proceedings of the 10th ACM SIGPLAN International Conference on Certified Programs and Proofs, co-located with POPL 2021. editor / Catalin Hritcu ; Andrei Popescu. Association for Computing Machinery, Inc, 2021. pp. 313-326 (CPP 2021 - Proceedings of the 10th ACM SIGPLAN International Conference on Certified Programs and Proofs, co-located with POPL 2021).

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title = "The generalised continuum hypothesis implies the axiom of choice in Coq",

abstract = "We discuss and compare two Coq mechanisations of Sierpinski's result that the generalised continuum hypothesis (GCH) implies the axiom of choice (AC). The first version shows the result, originally stated in first-order ZF set-theory, for a higher-order set theory convenient to work with in Coq. The second version presents a corresponding theorem for Coq's type theory itself, concerning type-theoretic formulations of GCH and AC. Both versions rely on the classical law of excluded middle and extensionality assumptions but we localise the use of axioms where possible.",

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author = "Dominik Kirst and Felix Rech",

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series = "CPP 2021 - Proceedings of the 10th ACM SIGPLAN International Conference on Certified Programs and Proofs, co-located with POPL 2021",

publisher = "Association for Computing Machinery, Inc",

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Kirst, D & Rech, F 2021, The generalised continuum hypothesis implies the axiom of choice in Coq. in C Hritcu & A Popescu (eds), CPP 2021 - Proceedings of the 10th ACM SIGPLAN International Conference on Certified Programs and Proofs, co-located with POPL 2021. CPP 2021 - Proceedings of the 10th ACM SIGPLAN International Conference on Certified Programs and Proofs, co-located with POPL 2021, Association for Computing Machinery, Inc, pp. 313-326, 10th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2021, co-located with POPL 2021, Virtual, Online, Denmark, 17/01/21. https://doi.org/10.1145/3437992.3439932

The generalised continuum hypothesis implies the axiom of choice in Coq. / Kirst, Dominik; Rech, Felix.
CPP 2021 - Proceedings of the 10th ACM SIGPLAN International Conference on Certified Programs and Proofs, co-located with POPL 2021. ed. / Catalin Hritcu; Andrei Popescu. Association for Computing Machinery, Inc, 2021. p. 313-326 (CPP 2021 - Proceedings of the 10th ACM SIGPLAN International Conference on Certified Programs and Proofs, co-located with POPL 2021).

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › peer-review

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AB - We discuss and compare two Coq mechanisations of Sierpinski's result that the generalised continuum hypothesis (GCH) implies the axiom of choice (AC). The first version shows the result, originally stated in first-order ZF set-theory, for a higher-order set theory convenient to work with in Coq. The second version presents a corresponding theorem for Coq's type theory itself, concerning type-theoretic formulations of GCH and AC. Both versions rely on the classical law of excluded middle and extensionality assumptions but we localise the use of axioms where possible.

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Kirst D, Rech F. The generalised continuum hypothesis implies the axiom of choice in Coq. In Hritcu C, Popescu A, editors, CPP 2021 - Proceedings of the 10th ACM SIGPLAN International Conference on Certified Programs and Proofs, co-located with POPL 2021. Association for Computing Machinery, Inc. 2021. p. 313-326. (CPP 2021 - Proceedings of the 10th ACM SIGPLAN International Conference on Certified Programs and Proofs, co-located with POPL 2021). doi: 10.1145/3437992.3439932

The generalised continuum hypothesis implies the axiom of choice in Coq

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Keywords

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