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INTERRELATION BETWEEN WEAK FRAGMENTS OF DOUBLE NEGATION SHIFT AND RELATED PRINCIPLES

  • MAKOTO FUJIWARA (a1) and ULRICH KOHLENBACH (a2)

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.

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[1]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. 192201.
[2]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. 337405.
[3]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. 6483.
[4]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. 704706.
[5]Dorais, F. G., Classical consequences of continuous choice principles from intuitionistic analysis. Notre Dame Journal of Formal Logic, vol. 55 (2014), no. 1, pp. 2539.
[6]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. 913971.
[7]Fujiwara, M., Intuitionistic and uniform provability in reverse mathematics, Ph.D. thesis, Tohoku University, 2015.
[8]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. 186195.
[9]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. 861870.
[10]Gödel, K., Über eine bisher noch nicht benützte Erweiterung des finiten Standpunktes. Dialectica, vol. 12 (1958), pp. 280287.
[11]Ishihara, H., Markov’s principle, Church’s thesis and Lindelöf’s theorem. Indagationes Mathematicae (N.S.), vol. 4 (1993), no. 3, pp. 321325.
[12]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. 245267.
[13]Kirby, L. and Paris, J., Accessible independence results for Peano arithmetic. Bulletin of the London Mathematical Society, vol. 14 (1982), no. 4, pp. 285293.
[14]Kohlenbach, U., Applied Proof Theory: Proof Interpretations and Their Use in Mathematics, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2008.
[15]Kohlenbach, U., On the disjunctive Markov principle. Studia Logica, vol. 103 (2015), no. 6, pp. 13131317.
[16]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. 379380.
[17]Kreisel, G. and Troelstra, A. S., Formal systems for some branches of intuitionistic analysis. Annals of Pure and Applied Logic, vol. 1 (1970), pp. 229387.
[18]Kuyper, R., On weihrauch reducibility and intuitionistic reverse mathematics, this JOURNAL, vol. 82 (2017), no. 4, pp. 14381458.
[19]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.
[20]Parsons, C., On n-quantifier induction, this JOURNAL, vol. 37 (1972), pp. 466482.
[21]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. 153160.
[22]Simpson, S. G., Subsystems of Second Order Arithmetic, second ed., Perspectives in Logic, Cambridge University Press, Cambridge, 2009.
[23]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. 127.
[24]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. 11881200.
[25]Troelstra, A. S., editor, Metamathematical Investigation of Intuitionistic Arithmetic and Analysis, Lecture Notes in Mathematics, vol. 344, Springer-Verlag, Berlin, New York, 1973.
[26]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.
[27]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.

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INTERRELATION BETWEEN WEAK FRAGMENTS OF DOUBLE NEGATION SHIFT AND RELATED PRINCIPLES

  • MAKOTO FUJIWARA (a1) and ULRICH KOHLENBACH (a2)

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