Abstract
Gödel [3] published a monograph in 1940 proving a highly significant theorem, namely that the axiom of choice (AC) and the generalized continuum hypothesis (GCH) are consistent with respect to the other axioms of set theory. This theorem addresses the first of Hilbert’s famous list of unsolved problems in mathematics. I have mechanized this work [8] using Isabelle/ZF [5,6]. Obviously, the theorem’s significance makes it a tempting challenge; the proof also has numerous interesting features. It is not a single formal assertion, as most theorems are. Gödel [3, p. 33] states it as follows, using Σ to denote the axioms for set theory:
What we shall prove is that, if a contradiction from the axiom of choice and the generalized continuum hypothesis were derived in Σ, it could be transformed into a contradiction obtained from the axioms of Σ alone.
Gödel presents no other statement of this theorem. Neither does he introduce a theory of syntax suitable for reasoning about transformations on proofs, surely because he considers it to be unnecessary
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
Bancerek, G., Rudnicki, P.: A compendium of continuous lattices in mizar. Journal of Automated Reasoning 29(3-4), 189–224 (2002)
Belinfante, J.G.F.: On computer-assisted proofs in ordinal number theory. Journal of Automated Reasoning 22(3), 341–378 (1999)
Gödel, K.: The consistency of the axiom of choice and of the generalized continuum hypothesis with the axioms of set theory. In: Feferman, S., et al. (eds.) Kurt Gödel: Collected Works, vol. II, pp. 33–101. Oxford University Press, Oxford (1990); First published in 1940 by Princeton University Press
Kunen, K.: Set Theory: An Introduction to Independence Proofs. North-Holland, Amsterdam (1980)
Paulson, L.C.: Set theory for verification: I. From foundations to functions. Journal of Automated Reasoning 11(3), 353–389 (1993)
Paulson, L.C.: Set theory for verification: II. Induction and recursion. Journal of Automated Reasoning 15(2), 167–215 (1995)
Paulson, L.C.: The reflection theorem: A study in meta-theoretic reasoning. In: Voronkov, A. (ed.) CADE 2002. LNCS (LNAI), vol. 2392, pp. 377–391. Springer, Heidelberg (2002)
Paulson, L.C.: The relative consistency of the axiom of choice — mechanized using Isabelle/ZF. LMS Journal of Computation and Mathematics 6, 198–248 (2003), http://www.lms.ac.uk/jcm/6/lms2003-001/
Quaife, A.: Automated deduction in von Neumann-Bernays-Gödel set theory. Journal of Automated Reasoning 8(1), 91–147 (1992)
Urban, J.: Basic facts about inaccessible and measurable cardinals. Journal of Formalized Mathematics 12 (2000), http://mizar.uwb.edu.pl/JFM/Vol12/card_fil.html
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 2008 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Paulson, L.C. (2008). The Relative Consistency of the Axiom of Choice — Mechanized Using Isabelle/ZF. In: Beckmann, A., Dimitracopoulos, C., Löwe, B. (eds) Logic and Theory of Algorithms. CiE 2008. Lecture Notes in Computer Science, vol 5028. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-69407-6_52
Download citation
DOI: https://doi.org/10.1007/978-3-540-69407-6_52
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-69405-2
Online ISBN: 978-3-540-69407-6
eBook Packages: Computer ScienceComputer Science (R0)