Hostname: page-component-586b7cd67f-t7fkt Total loading time: 0 Render date: 2024-11-20T08:43:19.540Z Has data issue: false hasContentIssue false
  • English
  • Français

INTERRELATION BETWEEN WEAK FRAGMENTS OF DOUBLE NEGATION SHIFT AND RELATED PRINCIPLES

Published online by Cambridge University Press:  23 October 2018

MAKOTO FUJIWARA  and
ULRICH KOHLENBACH
MAKOTO FUJIWARA
Affiliation:
WASEDA INSTITUTE FOR ADVANCED STUDY WASEDA UNIVERSITY 1-6-1 NISHI WASEDA, SHINJUKU-KU TOKYO 169-8050, JAPANE-mail: [email protected]
ULRICH KOHLENBACH
Affiliation:
FACHBEREICH MATHEMATIK TU DARMSTADT SCHLOSSGARTENSTRASSE 7 D-64289 DARMSTADT, GERMANYE-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We investigate two weak fragments of the double negation shift schema, which are motivated, respectively, from Spector’s consistency proof of ACA0 and from the negative translation of RCA0, as well as double negated variants of logical principles. Their interrelations over both intuitionistic arithmetic and analysis are completely solved.

Keywords

03B2003F5003F1003F25double negation shiftfragments of classical logicintuitionistic arithmetic and analysis
Type
Articles
Information
The Journal of Symbolic Logic , Volume 83 , Issue 3 , September 2018 , pp. 991 - 1012
Copyright
Copyright © The Association for Symbolic Logic 2018 

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 Computer Society, Washington, DC, 2004, pp. 192–201.Google Scholar
Avigad, J. and Feferman, S., Gödel’s functional (“Dialectica”) interpretation, Handbook of Proof Theory (Buss, S. R., editor), Studies in Logic and the Foundations of Mathematics, vol. 137, North-Holland, Amsterdam, 1998, pp. 337–405.CrossRefGoogle Scholar
Berardi, S. and Steila, S., Ramsey theorem for pairs as a classical principle in intuitionistic arithmetic, 19th International Conference on Types for Proofs and Programs (Matthes, R. and Schubert, A., editors), Leibniz International Proceedings in Informatics, vol. 26, Schloss Dagstuhl – Leibniz-Zentrum für Informatik GmbH, Wadern, 2014, pp. 64–83.Google Scholar
Cichon, E. A., A short proof of two recently discovered independence results using recursion theoretic methods. Proceedings of the American Mathematical Society, vol. 87 (1983), no. 4, pp. 704–706.CrossRefGoogle Scholar
Dorais, F. G., Classical consequences of continuous choice principles from intuitionistic analysis. Notre Dame Journal of Formal Logic, vol. 55 (2014), no. 1, pp. 25–39.CrossRefGoogle Scholar
Feferman, S., Theories of finite type related to mathematical practice, Handbook of Mathematical Logic (Barwise, J., editor), Studies in Logic and the Foundations of Mathematics, vol. 90, Elsevier, 1977, pp. 913–971.CrossRefGoogle Scholar
Fujiwara, M., Intuitionistic and uniform provability in reverse mathematics, Ph.D. thesis, Tohoku University, 2015.Google Scholar
Fujiwara, M., Intuitionistic provability versus uniform provability in $RCA$, Evolving Computability (Beckmann, A., Mitrana, V., and Soskova, M., editors), Lecture Notes in Computer Science, vol. 9136, Springer, Cham, 2015, pp. 186–195.CrossRefGoogle Scholar
Fujiwara, M., Ishihara, H., and Nemoto, T., Some principles weaker than Markov’s principle. Archive for Mathematical Logic, vol. 54 (2015), no. 7–8, pp. 861–870.CrossRefGoogle Scholar
Gödel, K., Über eine bisher noch nicht benützte Erweiterung des finiten Standpunktes. Dialectica, vol. 12 (1958), pp. 280–287.CrossRefGoogle Scholar
Ishihara, H., Markov’s principle, Church’s thesis and Lindelöf’s theorem. Indagationes Mathematicae (N.S.), 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 (Crosilla, L. and Schuster, P., editors), Oxford Logic Guides, vol. 48, Oxford University Press, Oxford, 2005, pp. 245–267.CrossRefGoogle Scholar
Kirby, L. and Paris, J., Accessible independence results for Peano arithmetic. Bulletin of the London Mathematical Society, vol. 14 (1982), no. 4, pp. 285–293.CrossRefGoogle Scholar
Kohlenbach, U., Applied Proof Theory: Proof Interpretations and Their Use in Mathematics, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2008.Google Scholar
Kohlenbach, U., On the disjunctive Markov principle. Studia Logica, vol. 103 (2015), no. 6, pp. 1313–1317.CrossRefGoogle Scholar
Kreisel, G., Proof theoretic results on intuitionistic first order arithmetic, abstract for the meeting of the Association for Symbolic Logic, Leeds 1962, this Journal, vol. 27 (1962), no. 3, pp. 379–380.Google Scholar
Kreisel, G. and Troelstra, A. S., Formal systems for some branches of intuitionistic analysis. Annals of Pure and Applied Logic, vol. 1 (1970), pp. 229–387.Google Scholar
Kuyper, R., On weihrauch reducibility and intuitionistic reverse mathematics, this JOURNAL, vol. 82 (2017), no. 4, pp. 1438–1458.Google Scholar
Moschovakis, J. R., Note on ${\rm{\Pi }}_{{\rm{n}} + 1}^0 - {\rm{LEM}}$, ${\rm{\Sigma }}_{{\rm{n}} + 1}^0 - {\rm{LEM}}$, ${\rm{\Pi }}_{{\rm{n}} + 1}^0 - {\rm{DNE}}$. Proceedings of 5th Panhellenic Logic Symposium, University of Athens, Athens, Greece, 2005.Google Scholar
Parsons, C., On n-quantifier induction, this JOURNAL, vol. 37 (1972), pp. 466–482.Google Scholar
Scedrov, A. and Vesley, R. E., On a weakening of Markov’s principle. Archive for Mathematical Logic Grundlagenforsch, vol. 23 (1983), no. 3–4, pp. 153–160.CrossRefGoogle Scholar
Simpson, S. G., Subsystems of Second Order Arithmetic, second ed., Perspectives in Logic, Cambridge University Press, Cambridge, 2009.CrossRefGoogle Scholar
Spector, C., Provably recursive functionals of analysis: A consistency proof of analysis by an extension of principles formulated in current intuitionistic mathematics, Proceedings of Symposia in Pure Mathematics, vol. V (Dekker, J. C. E., editor), American Mathematical Society, Providence, RI, 1962, pp. 1–27.Google Scholar
Toftdal, M., A calibration of ineffective theorems of analysis in a hierarchy of semi-classical logical principles (extended abstract), Automata, Languages and Programming (Díaz, J., Karhumäki, J., Lepistö, A., and Sannella, D., editors), Lecture Notes in Computer Science, vol. 3142, Springer, Berlin, 2004, pp. 1188–1200.CrossRefGoogle Scholar
Troelstra, A. S., editor, Metamathematical Investigation of Intuitionistic Arithmetic and Analysis, Lecture Notes in Mathematics, vol. 344, Springer-Verlag, Berlin, New York, 1973.CrossRefGoogle Scholar
Troelstra, A. S. and van Dalen, D., Constructivism in Mathematics, An Introduction, vol. I, Studies in Logic and the Foundations of Mathematics, vol. 121, North-Holland, Amsterdam, 1988. An introduction.Google Scholar
Troelstra, A. S. and van Dalen, D., Constructivism in Mathematics, An Introduction, vol. II, Studies in Logic and the Foundations of Mathematics, vol. 123, North-Holland, Amsterdam, 1988. An introduction.Google Scholar