Hostname: page-component-cd9895bd7-p9bg8 Total loading time: 0 Render date: 2024-12-26T07:57:52.487Z Has data issue: false hasContentIssue false
  • English
  • Français

INDEFINITENESS IN SEMI-INTUITIONISTIC SET THEORIES: ON A CONJECTURE OF FEFERMAN

Published online by Cambridge University Press:  29 June 2016

MICHAEL RATHJEN
MICHAEL RATHJEN*
Affiliation:
DEPARTMENT OF PURE MATHEMATICS UNIVERSITY OF LEEDS LEEDS LS2 9JT, UKE-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

The paper proves a conjecture of Solomon Feferman concerning the indefiniteness of the continuum hypothesis relative to a semi-intuitionistic set theory.

Keywords

MSC200003F5003F2503E5503B1503C70Continuum hypothesisindefinite conceptssemi-intuitionistic set theoryrealizabilityrelativized constructible hierarchyforcing
Type
Articles
Information
The Journal of Symbolic Logic , Volume 81 , Issue 2 , June 2016 , pp. 742 - 754
Copyright
Copyright © The Association for Symbolic Logic 2016 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Aczel, P. and Rathjen, M., Notes on constructive set theory, Technical Report 40, Institut Mittag-Leffler, The Royal Swedish Academy of Sciences, 2001, http://www.mittag-leffler.se/preprints/0001/, Preprint No. 40.Google Scholar
Aczel, P. and Rathjen, M., Notes on constructive set theory, preprint, 2010, 243 pages, http://www1.maths.leeds.ac.uk/rathjen/book.pdf.Google Scholar
Barwise, J., Admissible Sets and Structures, Springer, Berlin, 1975.Google Scholar
Beeson, M., Foundations of Constructive Mathematics, Springer-Verlag, Berlin, Heidelberg, New York, Tokyo, 1985.CrossRefGoogle Scholar
Chen, R.-M. and Rathjen, M., Lifschitz Realizability for Intuitionistic Zermelo-Fraenkel Set Theory . Archive for Mathematical Logic, vol. 51 (2012), pp. 789–818.Google Scholar
Devlin, K., Constructibility, Springer, Berlin, Heidelberg, New York, Tokyo, 1984.Google Scholar
Diaconescu, R., Axiom of choice and complementation . Proceedings of the American Mathematical Society, vol. 51 (1975), pp. 176–178.Google Scholar
Dummett, M., Frege: Philosophy of Mathematics, Harvard University Press, London, 1991.Google Scholar
Feferman, S., On the strength of some semi-constructive theories , Logic, Construction, Computation (Berger, U., Schuster, P., and Seisenberger, M., editors), pp. 201–225, Ontos Verlag, Frankfurt, 2012.Google Scholar
Feferman, S., Is the continuum hypothesis a definite mathematical problem?, Draft of paper for the lecture to the Philosophy Dept., Harvard University, Oct. 5, 2011 in the Exploring the Frontiers of Incompleteness project series, Havard 2011–2012.Google Scholar
Feferman, S., Three Problems for Mathematics: Lecture 2: Is the Continuum Hypothesis a definite mathematical problem? Slides for inaugural Paul Bernays Lectures, ETH, Zürich, Sept. 12, 2012.Google Scholar
Feferman, S., Why isn’t the Continuum Problem on the Millennium ($1,000,000) Prize list? Slides for CSLI Workshop on Logic, Rationality and Intelligent Interaction, Stanford, June 1, 2013.Google Scholar
Frege, G., Die Grundlagen der Arithmetik, Verlag Wilhelm Koebner, Breslau, 1884.Google Scholar
Friedman, H., Some applications of Kleene’s method for intuitionistic systems , Cambridge Summer School in Mathematical Logic (Mathias, A. and Rogers, H., editors), Lectures Notes in Mathematics, vol. 337, Springer, Berlin, 1973, pp. 113–170.Google Scholar
Hajnal, A., On a Consistency Theorem Connected with the Generalised Continuum Problem . Zeitschrift für Mathematische Logik, vol. 2 (1956), pp. 131–136.Google Scholar
Jech, T., Set Theory, Springer, Berlin, 2003.Google Scholar
Kunen, K., Set Theory, North-Holland, Amsterdam, New York, Oxford, 1980.Google Scholar
Lévy, A., 1957 Indépendence Conditionnelle de V = L et d’Axiomes qui se Rattachent au Systeme de M. Gödel , Comptes Rendus de l’Académie des Sciences, Paris, vol. 245 (1957), pp. 1582–1583.Google Scholar
McCarty, D. C., Realizability and recursive set theory . Annals of Pure and Applied Logic, vol. 32 (1986), pp. 153–183.Google Scholar
Myhill, J., Some properties of Intuitionistic Zermelo-Fraenkel set theory , Cambridge Summer School in Mathematical Logic (Mathias, A. and Rogers, H., editors), vol. 337, Lectures Notes in Mathematics, Springer, Berlin, 1973, pp. 206–231.Google Scholar
Pozsgay, L., Liberal intuitionism as a basis for set theory , Axiomatic Set Theory, Proceedings of Symposia in Pure Mathematics XIII, vol. 1 (1971), pp. 321–330.Google Scholar
Pozsgay, L., Semi-intuitionistic set theory . Notre Dame Journal of Formal Logic, vol. 13 (1972), pp. 546–550.Google Scholar
Rathjen, M., Realizability for Constructive Zermelo-Fraenkel Set Theory (Väänänen, J. and Stoltenberg-Hansen, V., editors), Logic Colloquium 2003, Lecture Notes in Logic 24, A.K. Peters, Natick, MA, 2006, pp. 282–314.Google Scholar
Rathjen, M., The formulae-as-classes interpretation of constructive set theory, Proof Technology and Computation (Schwichtenberg, H. and Spies, K., editors), pp. 279–322, IOS Press, Amsterdam, 2006.Google Scholar
Rathjen, M., The disjunction and other properties for constructive Zermelo-Fraenkel set theory, this Journal, vol. 70 (2005), pp. 1233–1254.Google Scholar
Rathjen, M., Metamathematical Properties of Intuitionistic Set Theories with Choice Principles , New Computational Paradigms: Changing Conceptions of What is Computable, (Cooper, S. B., Löwe, B., Sorbi, A., editors), Springer, New York, 2008, pp. 287–312.Google Scholar
Rathjen, M., From the weak to the strong existence property . Annals of Pure and Applied Logic, vol. 163 (2012), pp. 1400–1418.Google Scholar
Takeuti, G. and Wilson, W. M., Introduction to Axiomatic Set Theory, Springer, New York, Heidelberg, Berlin, 1971.Google Scholar
Tharp, L., A quasi-intuitionistic set theory, this Journal, vol. 36 (1971), pp. 456–460.Google Scholar
Wolf, R. S., Formally Intuitionistic Set Theories with Bounded Predicates Decidable, PhD Thesis, Stanford University, Stanford, CA, 1974.Google Scholar