Hostname: page-component-586b7cd67f-r5fsc Total loading time: 0 Render date: 2024-11-26T04:06:53.805Z Has data issue: false hasContentIssue false
  • English
  • Français

SEPARATING THE FAN THEOREM AND ITS WEAKENINGS II

Published online by Cambridge University Press:  25 July 2019

ROBERT S. LUBARSKY
ROBERT S. LUBARSKY*
Affiliation:
DEPARTMENT OF MATHEMATICAL SCIENCES FLORIDA ATLANTIC UNIVERSITY BOCA RATON, FL33431, USA E-mail:[email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

Varieties of the Fan Theorem have recently been developed in reverse constructive mathematics, corresponding to different continuity principles. They form a natural implicational hierarchy. Earlier work showed all of these implications to be strict. Here we reprove one of the strictness results, using very different arguments. The technique used is a mixture of realizability, forcing in the guise of Heyting-valued models, and Kripke models.

Keywords

03C9003E7003F5003F6003D80Fan theoremsKripke modelsforcingHeyting-valued modelsformal topologyrecursive realizability
Type
Articles
Information
The Journal of Symbolic Logic , Volume 84 , Issue 4 , December 2019 , pp. 1484 - 1509
Copyright
Copyright © The Association for Symbolic Logic 2019 

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

Beeson, M., Foundations of Constructive Mathematics, Springer-Verlag, Berlin, 1985.10.1007/978-3-642-68952-9CrossRefGoogle Scholar
Berger, J., The logical strength of the uniform continuity theorem, Logical Approaches to Computational Barriers, 2006, Proceedings of CiE 2006, LNCS 3988, pp. 35–39.10.1007/11780342_4CrossRefGoogle Scholar
Berger, J., A separation result for varieties of Brouwer’s fan theorem. Proceedings of the 10th Asian Logic Conference (ALC 10), Kobe University in Kobe, Hyogo, Japan, September 1–6, 2008, 2010, pp. 85–92.Google Scholar
Diener, H. and Loeb, I., Sequences of real functions on [0, 1] in constructive reverse mathematics. Annals of Pure and Applied Logic, vol. 157 (2009), no. 1, pp. 50–61.10.1016/j.apal.2008.09.018CrossRefGoogle Scholar
Diener, H. and Lubarsky, R., Notions of cauchyness and metastability, Logical Foundations in Computer Science (Artemov, S. and Nerode, A., editors), Lecture Notes in Computer Science 10703, Springer, Berlin/Heidelberg, 2018.Google Scholar
Fourman, M. and Hyland, J., Sheaf models for analysis, Applications of Sheaves (Fourman, M., Mulvey, C., and Scott, D., editors), Lecture Notes in Mathematics, vol. 753, Springer, Berlin/Heidelberg, 1979, pp. 280–301.CrossRefGoogle Scholar
Fourman, M. and Scott, D., Sheaves and logic, Applications of Sheaves (Fourman, M., Mulvey, C., and Scott, D., editors), Lecture Notes in Mathematics, vol. 753, Springer, Berlin/Heidelberg, 1979, pp. 302–401.CrossRefGoogle Scholar
Friedman, H. and Scedrov, A., The lack of definable witnesses and provably recursive functions in intuitionistic set theories. Advances in Math, (1985), pp. 1–13.10.1016/0001-8708(85)90103-3CrossRefGoogle Scholar
Gambino, N., Heyting-valued interpretations for constructive set theory. Annals of Pure and Applied Logic, vol. 137 (2006), pp. 164–188.CrossRefGoogle Scholar
Julian, W. and Richman, F., A uniformly continuous function on [0,1] that is everywhere difierent from its infimum. Pacific Journal of Mathematics, vol. 111 (1984), no. 2, pp. 333–340.CrossRefGoogle Scholar
Lubarsky, R., Intuitionistic L, Logical Methods in Computer Science: The Nerode Conference (Crossley, et al., editors), Birkhauser, 1993, pp. 555–571.Google Scholar
Lubarsky, R., Independence results around constructive ZF. Annals of Pure and Applied Logic, vol. 132 (2005), pp. 209–225.CrossRefGoogle Scholar
Lubarsky, R., Separating the fan theorem and its weakenings II, Logical Foundations of Computer Science (Artemov, S. and Nerode, A., editors), Lecture Notes in Computer Science 10703, Springer, Berlin/Heidelberg, 2018, pp. 242–255.10.1007/978-3-319-72056-2_15CrossRefGoogle Scholar
Lubarsky, R. and Diener, H., Separating the fan theorem and its weakenings, this Journal, vol. 79 (2014), pp. 792–813.Google Scholar
McCarty, D., Realizability and recursive mathematics, Ph.D. thesis, Oxford University, and Carnegie-Mellon University, Technical report CMU-CS-84-131, 1984.Google Scholar
McCarty, D., Realizability and recursive set theory. Annals of Pure and Applied Logic, vol. 32 (1986), pp. 153–183.CrossRefGoogle Scholar
van Oosten, J, Realizability: An Introduction to its Categorical Side, Elsevier, Amsterdam, 2008.Google Scholar
Troelstra, A. S. and van Dalen, D., Constructivism in Mathematics, vol. 2, Studies in Logic, vol. 123, Elsevier, Amsterdam, 1988.Google Scholar