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The Worm principle

Published online by Cambridge University Press:  31 March 2017

Zoé Chatzidakis
Affiliation:
Université de Paris VII (Denis Diderot)
Peter Koepke
Affiliation:
Rheinische Friedrich-Wilhelms-Universität Bonn
Wolfram Pohlers
Affiliation:
Westfälische Wilhelms-Universität Münster, Germany
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Publisher: Cambridge University Press
Print publication year: 2006

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References

[1] W., Ackermann, Zur Widerspruchsfreiheit der Zahlentheorie Mathematische Annalen, vol. 117 (1940), pp. 162–194.
[2] L. D., Beklemishev, Induction rules, reflection principles, and provably recursive functions Annals of Pure and Applied Logic, vol. 85 (1997), no. 3, pp. 193–242.
[3] L. D., Beklemishev, Proof-theoretic analysis by iterated reflection Archive for Mathematical Logic, vol. 42 (2003), no. 6, pp. 515–552, DOI: 10.1007/s00153-002-0158-7.
[4] L. D., Beklemishev, Provability algebras and proof-theoretic ordinals. I Annals of Pure and Applied Logic, vol. 128 (2004), no. 1-3, pp. 103–123.
[5] G., Boolos, The Logic of Provability, Cambridge University Press, Cambridge, 1993.
[6] W., Buchholz, An independence result for (Π1/1-CA)+BI, Annals of Pure and Applied Logic, vol. 33 (1987), no. 2, pp. 131–155.
[7] W., Buchholz and S., Wainer, Provably computable functions and the fast growing hierarchy Logic and Combinatorics (Arcata, Calif., 1985), Contemporary Mathematics, vol. 65, AMS, Providence, RI, 1987, pp. 179–198.
[8] G. K., Dzhaparidze, Polymodal provability logic Intensional Logics and the Logical Structure of Theories (Telavi, 1985), “Metsniereba”, Tbilisi, 1988, (Russian), pp. 16–48.
[9] A., Grzegorczyk, Some classes of recursive functions Rozprawy Matematyczne, vol. 4 (1953), p. 46.
[10] M., Hamano and M., Okada, A relationship among Gentzen's proof-reduction, Kirby-Paris' hydra game and Buchholz's hydra game Mathematical Logic Quarterly, vol. 43 (1997), no. 1, pp. 103–120.
[11] L. A. S., Kirby and J. B., Paris, Accessible independence results for Peano arithmetic The Bulletin of the London Mathematical Society, vol. 14 (1982), no. 4, pp. 285–293.
[12] G., Kreisel, On the interpretation of non-finitist proofs. II. Interpretation of number theory. Applications The Journal of Symbolic Logic, vol. 17 (1952), pp. 43–58.
[13] G., Kreisel, Wie die Beweistheorie zu ihren Ordinalzahlen kam und kommt Jahresbericht der Deutschen Mathematiker-Vereinigung, vol. 78 (1976/77), no. 4, pp. 177–223.
[14] G., Kreisel and A., Lévy, Reflection principles and their use for establishing the complexity of axiomatic systems Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, vol.14 (1968), pp. 97–142.
[15] D., Leivant, The optimality of induction as an axiomatization of arithmetic The Journal of Symbolic Logic, vol. 48 (1983), no. 1, pp. 182–184.
[16] G. E., Minc, Quantifier-free and one-quantifier systems Zapiski Naučnyh Seminarov Leningradskogo Otdelenija Matematičeskogo Instituta im. V. A. Steklova Akademii Nauk SSSR (LOMI), vol. 20 (1971), pp. 115–133, In Russian.
[17] C., Parsons, On a number theoretic choice schema and its relation to induction Intuitionism and Proof Theory (Proc. Conf., Buffalo, N.Y., 1968) (A., Kino, J., Myhill, and R. E., Vessley, editors), North-Holland, Amsterdam, 1970, pp. 459–473.
[18] H. E., Rose, Subrecursion: Functions and Hierarchies, Oxford Logic Guides, vol. 9, The Clarendon Press, Oxford University Press, New York, 1984.

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