Hostname: page-component-78c5997874-g7gxr Total loading time: 0 Render date: 2024-11-17T19:16:09.316Z Has data issue: false hasContentIssue false
  • English
  • Français

SEPARATING FRAGMENTS OF WLEM, LPO, AND MP

Published online by Cambridge University Press:  01 December 2016

MATT HENDTLASS  and
ROBERT LUBARSKY
MATT HENDTLASS
Affiliation:
DEPARTMENT OF MATHEMATICS AND STATISTICS UNIVERSITY OF CANTERBURY PRIVATE BAG 4800, CHRISTCHURCH, NEW ZEALANDE-mail: m.hendtlass@gmail.com
ROBERT LUBARSKY
Affiliation:
DEPARTMENT OF MATHEMATICAL SCIENCES FLORIDA ATLANTIC UNIVERSITY BOCA RATON, FL 33431, USAE-mail: rlubarsk@fau.edu
Get access Rights & Permissions [Opens in a new window]

Abstract

We separate many of the basic fragments of classical logic which are used in reverse constructive mathematics. A group of related Kripke and topological models is used to show that various fragments of the Weak Law of the Excluded Middle, the Limited Principle of Omniscience, and Markov’s Principle, including Weak Markov’s Principle, do not imply each other.

Keywords

03F5003C9003E7003H99omniscience principlesMarkov’s PrincipleWeak Markov’s Principleweak LEMtopological modelsKripke modelsnonstandard models
Type
Articles
Information
The Journal of Symbolic Logic , Volume 81 , Issue 4 , December 2016 , pp. 1315 - 1343
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

Akama, Y., Berardi, S., Hayashi, S., and Kohlenbach, U., An Arithmetical Hierarchy of the Law of Excluded Middle and Related Principles, Proceedings of the 19th Annual IEEE Symposium on Logic in Computer Science (LICS ’04), IEEE Press, New Jersey, 2004, pp. 192–201.Google Scholar
Bishop, E. A., Foundations of Constructive Analysis, McGraw-Hill, New York, 1967.Google Scholar
Brattka, V. and Gherardi, G., Weihrauch degrees, omniscience principles and weak computability , this Journal, vol. 76 (2011), no. 1, pp. 143–176.Google Scholar
Brattka, V., Hendtlass, M., and Kreuzer, A. P., On the uniform computational content of computability theory, http://arxiv.org/abs/1501.00433.Google Scholar
Chen, R.-M. and Rathjen, M., Lifschitz realizability for intuitionistic Zermelo-Fraenkel set theory . Archive for Mathematical Logic, vol. 51 (2012), no. 7, 8, pp. 789–818.CrossRefGoogle Scholar
van Dalen, D., An interpretation of intuitionistic analysis . Annals of Mathematical Logic, vol. 13 (1978), no. 1, pp. 1–43.CrossRefGoogle Scholar
Friedman, H. M., Set theoretic foundations for constructive analysis . Annals of Mathematics, vol. 105 (1977), no. 1, pp. 1–28.Google Scholar
Friedman, H. and Scedrov, A., The lack of definable witnesses and provably recursive functions in intuitionistic set theories . Advances in Mathematics, vol. 57 (1985), pp. 1–13.Google Scholar
Grayson, R. J., Heyting-valued semantics , Logic Colloquium ’82 (Lolli, G., Longo, G., and Marcja, A., editors), Studies in Logic and the Foundations of Mathematics 112, North Holland, Amsterdam, 1984, pp. 181–208.Google Scholar
Ishihara, H., Markov’s principle, Church’s thesis and Lindelöf’s theorem. Indagationes Mathematicae, vol. 4 (1993), no. 3, pp. 321–325.CrossRefGoogle Scholar
Ishihara, H., Constructive reverse mathematics: Compactness properties , From Sets and Types to Topology and Analysis: Towards Practicable Foundations for Constructive Mathematics (Crosilla, L. and Schuster, P., editors), Oxford University Press, Oxford, 2005, pp. 245–267.Google Scholar
Kohlenbach, U., Relative constructivity , this Journal, vol. 63 (1998), pp. 1218–1238.Google Scholar
Kohlenbach, U., On weak Markov’s principle . Mathematical Logic Quarterly, vol. 48 (2002), Suppl. 1, pp. 59–65.Google Scholar
Kohlenbach, U., Applied Proof Theory: Proof Interpretations and their Use in Mathematics, Springer, Berlin, 2008.Google Scholar
Krol, M., A topological model for intuitionistic analysis with Kripke’s Scheme . Zeitschrift für mathematische Logik und Grundlagen der Mathematik, vol. 24 (1978), pp. 427–436.CrossRefGoogle Scholar
Lubarsky, R. and Rathjen, M., Realizability models separating various fan theorems , The Nature of Computation (Bonnizoni, P., Brattka, V., and Löwe, B., editors), Proceedings of Computability in Europe 2013, LNCS 7921, Springer, Heidelberg, 2013, pp. 306–315.Google Scholar
Mandelkern, M., Constructively complete finite sets . Zeitschrift für mathematische Logik und Grundlagen der Mathematik, vol. 34 (1988), pp. 97–103.CrossRefGoogle Scholar
[18] Mylatz, U., Vergleich unstetiger Funktionen: “Principle of Omniscience” und Vollständigkeit in der C-Hierarchie, Ph.D. dissertation, Fernuniversität Hagen, 2006.Google Scholar
Moschovakis, J., A topological interpretation of second-order intuitionistic arithmetic . Compositio Mathematica, vol. 26 (1973), pp. 261–276.Google Scholar
van Oosten, J., Realizability, Studies in Logic and the Foundations of Mathematics 152, Elsevier, Amsterdam, 2008.Google Scholar
Rathjen, M., Constructive Zermelo-Fraenkel set theory and the limited principle of omniscience . Annals of Pure and Applied Logic, vol. 165 (2014), pp. 563–572.Google Scholar
Richman, F., Polynomials and linear transformations . Linear Algebra and its Applications, vol. 131 (1990), pp. 131–137.Google Scholar
Richman, F., Weak Markov’s principle, strong extensionality, and countable choice, unpublished manuscript, 2000, available at http://math.fau.edu/richman/HTML/DOCS.HTM.Google Scholar
Scott, D., Identity and existence in intuitionistic logic , Applications of Sheaves (Fourman, M., Mulvey, C., and Scott, D., editors), Lecture Notes in Mathematics 753, Springer, Berlin, 1979, pp. 660–696.Google Scholar
Simpson, S., Subsystems of Second Order Arithmetic, Association for Symbolic Logic/Cambridge University Press, Cambridge, 2009.CrossRefGoogle Scholar
Toftdal, M., Calibration of ineffective theorems of analysis in a constructive context , Master’s thesis, University of Aarhus, 2004; also as A Calibration of ineffective theorems of analysis in a hierarchy of semi-classical logical principles, ICALP 2004, LNCS 3142, 2004, pp. 1188–1200.Google Scholar
Troelstra, A. S., Metamathematical Investigation of Intuitionistic Arithmetic and Analysis, Lecture Notes in Mathematics 344, Springer, Berlin, 1973.Google Scholar
Troelstra, A. S., Realizability , Handbook of Proof Theory (Buss, S., editor), Studies in Logic and the Foundations of Mathematics 137, Elsevier, Amsterdam, 1998, pp. 407–474.Google Scholar
Troelstra, A. S. and van Dalen, D., Constructivism in Mathematics, vol. 1, Studies in Logic and the Foundations of Mathematics 121, North Holland, Amsterdam, 1988.Google Scholar
Troelstra, A. S. and van Dalen, D., Constructivism in Mathematics, vol. 2, Studies in Logic and the Foundations of Mathematics 123, North Holland, Amsterdam, 1988.Google Scholar
Vesley, R. E., Realizing Brouwer’s sequences . Annals of Pure and Applied Logic, vol. 81 (1996), pp. 25–74.CrossRefGoogle Scholar
[32] Weihrauch, K., The TTE-Interpretation of Three Hierarchies of Omniscience Principles , Informatik Berichte Nr. 130, Fernuniversität Hagen, Hagen, 1992.Google Scholar
Yu, X. and Simpson, S. G., Measure theory and weak König’s lemma . Archive for Mathematical Logic, vol. 30 (1990), pp. 171–180.Google Scholar