Skip to main content Accessibility help

We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.

Close cookie message
×
Home
Hostname: page-component-55597f9d44-dfw9g Total loading time: 0.243 Render date: 2022-08-13T06:18:20.561Z Has data issue: true Feature Flags: { "shouldUseShareProductTool": true, "shouldUseHypothesis": true, "isUnsiloEnabled": true, "useRatesEcommerce": false, "useNewApi": true } hasContentIssue true
  • English
LMS Journal of Computation and Mathematics LMS Journal of Computation and Mathematics

Article contents

The Relative Consistency of the Axiom of Choice Mechanized Using Isabelle⁄zf

Published online by Cambridge University Press:  01 February 2010

Lawrence C. Paulson
Lawrence C. Paulson
Affiliation:
Computer Laboratory, University of Cambridge, JJ Thomson Avenue, Cambridge CB3 OFD, Englandlcp@cl.cam.ac.uk, http://www.cl.cam.ac.uk/users/lcp

Abstract

HTML view is not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

The proof of the relative consistency of the axiom of choice has been mechanized using Isabelle⁄ZF, building on a previous mechanization of the reflection theorem. The heavy reliance on metatheory in the original proof makes the formalization unusually long, and not entirely satisfactory: two parts of the proof do not fit together. It seems impossible to solve these problems without formalizing the metatheory. However, the present development follows a standard textbook, Kenneth Kunen's Set theory: an introduction to independence proofs, and could support the formalization of further material from that book. It also serves as an example of what to expect when deep mathematics is formalized.

Type
Research Article
Information
LMS Journal of Computation and Mathematics , Volume 6 , 2003 , pp. 198 - 248
Copyright
Copyright © London Mathematical Society 2003

References

1Bruijn, N.G. De, ‘Lambda calculus notation with nameless dummies, a tool for automatic formula manipulation, with application to the Church-Rosser theorem’, Indag. Math. 34 (1972) 381–392.CrossRefGoogle Scholar
2Davey, B.A. and Priestley, H.A., Introduction to lattices and order (Cambridge University Press, 1990).Google Scholar
3Feferman, S. et al. , eds, Kurt Gödel: collected works, vol. II. (Oxford University Press, 1990).Google Scholar
4Gödel, Kurt, ‘The consistency of the axiom of choice and of the generalized continuum hypothesis’, [3] 26–27; first published in Proc. Nat. Acad. Sci. USA (1938) 556–557.Google Scholar
5Gödel, Kurt, ‘The consistency of the axiom of choice and of the generalized continuum hypothesis with the axioms of set theory’, [3] 33–101; first published by Princeton University Press, 1940.Google Scholar
6Gödel, Kurt, ‘Consistency proof for the generalized continuum hypothesis’, [3] 27–32; first published in Proc. Nat. Acad. Sci. USA (1939) 220–224.Google Scholar
7Halmos, Paul R., Naive set theory (Van Nostrand, 1960).Google Scholar
8Kammüller, Florian, Wenzel, Markus and Paulson, Lawrence C. ‘Locales: a sectioning concept for Isabelle’, Theorem proving in higher order logics: TPHOLs '99, Lecture Notes in Comput. Sci. 1690 (ed. Bertot, Yves, Dowek, Gilles, Hirschowitz, André, Paulin, Christine and Théry, Laurent, Springer, 1999) 149–165.CrossRefGoogle Scholar
9Kunen, Kenneth, Set theory: an introduction to independence proofs (North-Holland, 1980).Google Scholar
10Mendelson, E., Introduction to mathematical logic, 4th edn (Chapman and Hall, 1997).Google Scholar
11Nipkow, Tobias, Paulson, Lawrence C. and Wenzel, Markus, Isabelle⁄HOL: a proof assistant for higher-order logic, Lecture Notes in Comput. Sci. Tutorial 2283 (Springer, 2002).CrossRefGoogle Scholar
12Paulson, Lawrence C., ‘The foundation of a generic theorem prover‘, J. Automat. Reasoning 5 (1989) 363–397.CrossRefGoogle Scholar
13Paulson, Lawrence C., ‘Set theory for verification: I. From foundations to functions‘, J. Automat. Reasoning 11 (1993) 353–389.CrossRefGoogle Scholar
14Paulson, Lawrence C., Isabelle: a generic theorem prover, Lecture Notes in Comput. Sci. 828 (Springer, 1994).Google Scholar
15Paulson, Lawrence C., ‘Set theory for verification: II. Induction and recursion’ J. Automat. Reasoning 15 (1995) 167–215.CrossRefGoogle Scholar
16Paulson, Lawrence C., ‘Proving properties of security protocols by induction‘, 10th Computer Security Foundations Workshop (IEEE Computer Society Press, 1997) 70–83.CrossRefGoogle Scholar
17Paulson, Lawrence C., ‘A fixedpoint approach to (co)inductive and (co)datatype definitions’, Proof, language, and interaction: essays in honor of Robin Milner (ed. Plotkin, Gordon, Stirling, Colin, and Tofte, Mads, MIT Press, 2000) 187–211.Google Scholar
18Paulson, Lawrence C., ‘The reflection theorem: a study in meta-theoretic reasoning’, [22] 377–391.Google Scholar
19Paulson, Lawrence C. and Grąbczewski, Krzysztof, ‘Mechanizing set theory:cardinal arithmetic and the axiom of choice’, J. Automat. Reasoning 17 (1996) 291–323.Google Scholar
20Prawitz, Dag, ‘Ideas and results in proof theory‘, Second Scandinavian Logic Symposium (ed. Fenstad, J.E., North-Holland, 1971) 235’308.Google Scholar
21Strecker, Martin, ‘Formal verification of a Java compiler in Isabelle’, [22] 63–77.Google Scholar
22Voronkov, Andrei, ed. Automated deduction – CADE-18 International Conference, Lecture Notes in Artificial Intelligence 2392 (Springer, 2002).CrossRefGoogle Scholar
Supplementary material: PDF

JCM 6 Paulson Appendix A

Paulson Appendix A

Download JCM 6 Paulson Appendix A(PDF)
PDF 749 KB
You have Access
11
Cited by

Save article to Kindle

To save this article to your Kindle, first ensure coreplatform@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.

Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

The Relative Consistency of the Axiom of Choice Mechanized Using Isabelle⁄zf
Available formats
×

Save article to Dropbox

To save this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you used this feature, you will be asked to authorise Cambridge Core to connect with your Dropbox account. Find out more about saving content to Dropbox.

The Relative Consistency of the Axiom of Choice Mechanized Using Isabelle⁄zf
Available formats
×

Save article to Google Drive

To save this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you used this feature, you will be asked to authorise Cambridge Core to connect with your Google Drive account. Find out more about saving content to Google Drive.

The Relative Consistency of the Axiom of Choice Mechanized Using Isabelle⁄zf
Available formats
×
×

Reply to: Submit a response

Please enter your response.

Your details

Please enter a valid email address.

Conflicting interests

Do you have any conflicting interests? *