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    This article has been cited by the following publications. This list is generated based on data provided by CrossRef.
    Paulson, Lawrence C. 2018. Computational logic: its origins and applications. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Science, Vol. 474, Issue. 2210, p. 20170872.
    Paulson, Lawrence C. 2015. A Mechanised Proof of Gödel’s Incompleteness Theorems Using Nominal Isabelle. Journal of Automated Reasoning, Vol. 55, Issue. 1, p. 1.
    Harrison, John Urban, Josef and Wiedijk, Freek 2014. Computational Logic. Vol. 9, Issue. , p. 135.
    Wenzel, Makarius Paulson, Lawrence C. and Nipkow, Tobias 2008. Theorem Proving in Higher Order Logics. Vol. 5170, Issue. , p. 33.
    Manolios, Panagiotis and Vroon, Daron 2005. Ordinal Arithmetic: Algorithms and Mechanization. Journal of Automated Reasoning, Vol. 34, Issue. 4, p. 387.
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The Relative Consistency of the Axiom of Choice Mechanized Using Isabelle⁄zf

  • Lawrence C. Paulson (a1)
Abstract

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.

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Copyright
COPYRIGHT: © London Mathematical Society 2003
References
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2Davey, B.A. and Priestley, H.A., Introduction to lattices and order (Cambridge University Press, 1990).
3Feferman, S. et al. , eds, Kurt Gödel: collected works, vol. II. (Oxford University Press, 1990).
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.
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.
6Gödel, Kurt, ‘Consistency proof for the generalized continuum hypothesis’, [3] 27–32; first published in Proc. Nat. Acad. Sci. USA (1939) 220–224.
7Halmos, Paul R., Naive set theory (Van Nostrand, 1960).
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.
9Kunen, Kenneth, Set theory: an introduction to independence proofs (North-Holland, 1980).
10Mendelson, E., Introduction to mathematical logic, 4th edn (Chapman and Hall, 1997).
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).
12Paulson, Lawrence C., ‘The foundation of a generic theorem prover‘, J. Automat. Reasoning 5 (1989) 363–397.
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14Paulson, Lawrence C., Isabelle: a generic theorem prover, Lecture Notes in Comput. Sci. 828 (Springer, 1994).
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16Paulson, Lawrence C., ‘Proving properties of security protocols by induction‘, 10th Computer Security Foundations Workshop (IEEE Computer Society Press, 1997) 70–83.
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.
18Paulson, Lawrence C., ‘The reflection theorem: a study in meta-theoretic reasoning’, [22] 377–391.
19Paulson, Lawrence C. and Grąbczewski, Krzysztof, ‘Mechanizing set theory:cardinal arithmetic and the axiom of choice’, J. Automat. Reasoning 17 (1996) 291–323.
20Prawitz, Dag, ‘Ideas and results in proof theory‘, Second Scandinavian Logic Symposium (ed. Fenstad, J.E., North-Holland, 1971) 235’308.
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22Voronkov, Andrei, ed. Automated deduction – CADE-18 International Conference, Lecture Notes in Artificial Intelligence 2392 (Springer, 2002).
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LMS Journal of Computation and Mathematics
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