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Models of transfinite provability logic

Published online by Cambridge University Press:  12 March 2014

David Fernández-Duque  and
Joost J. Joosten
David Fernández-Duque
Affiliation:
Department of Computer Science and Artificial Intelligence, Universidad de Sevilla, Av. Reina Mercedes S/N, 41012 Seville, Spain, E-mail:dfduque@us.es
Joost J. Joosten
Affiliation:
Department of Logic, History and Philosophy of Science, University of Barcelona, C. Montalegre, 6, 08001 Barcelona, Spain, E-mail:jjoosten@ub.edu
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Abstract

For any ordinal Λ, we can define a polymodal logic GLPΛ, with a modality [ξ] for each ξ < Λ. These represent provability predicates of increasing strength. Although GLPΛ has no Kripke models, Ignatiev showed that indeed one can construct a Kripke model of the variable-free fragment with natural number modalities, denoted . Later, Icard defined a topological model for which is very closely related to Ignatiev's.

In this paper we show how to extend these constructions for arbitrary Λ. More generally, for each Θ, Λ we build a Kripke model and a topological model , and show that is sound for both of these structures, as well as complete, provided Θ is large enough.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 78 , Issue 2 , June 2013 , pp. 543 - 561
Copyright
Copyright © Association for Symbolic Logic 2013

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References

REFERENCES

[1] Beklemishev, L. D., Fernández-Duque, D., and Joosten, J. J., On provability logics with linearly ordered modalities, forthcoming, 2012.Google Scholar
[2] Beklemishev, L. D. and Gabelaia, D., Topological completeness of the provability logicGLP, ArXiv, (2011), pp. 1–26.Google Scholar
[3] Beklemishev, L.D., Provability algebras and proof-theoretic ordinals, I, Annals of Pure and Applied Logic, vol. 128 (2004), pp. 103–124.CrossRefGoogle Scholar
[4] Beklemishev, L.D., Joosten, J.J., and Vervoort, M., A finitary treatment of the closed fragment of Japaridze's provability logic. Journal of Logic and Computation, vol. 15 (2005), pp. 447–463.CrossRefGoogle Scholar
[5] Fernández-Duque, D. and Joosten, J. J., Hyperations, Vehlen progressions and transfinite iteration of ordinal functions, submitted, 05 2012.Google Scholar
[6] Fernández-Duque, D., Kripke models of transfinite provability logic, Advances in modal logic, vol. 9, College Publications, 2012, pp. 185–199.Google Scholar
[7] Fernández-Duque, D., Turing progressions and their well-orders. How the world computes. Lecture Notes in Computer Science, Springer, 2012, pp. 212–221.CrossRefGoogle Scholar
[8] Fernández-Duque, D., Well-orders in the transfinite Japaridze algebra, submitted, 03 2012.Google Scholar
[9] Fernández-Duque, D., Well-orders in the transfinite Japaridze algebra II: Turing progressions and their well-orders, forthcoming, 2012.Google Scholar
[10] Icard, T. F. III, A topological study of the closed fragment of GLP, Journal of Logic and Computation, vol. 21 (2011), pp. 683–696.CrossRefGoogle Scholar
[11] Ignatiev, K.N., On strong provability predicates and the associated modal logics, this Journal, vol. 58 (1993), pp. 249–290.Google Scholar
[12] Japaridze, G., The polymodal provability logic, Intensional logics and logical structure of theories: material from the Fourth Soviet-Finnish symposium on logic, Metsniereba, Telavi, 1988, In Russian.Google Scholar
[13] Joosten, J. J., lntepretability formalized. Department of Philosophy, University of Utrecht, 2004, Ph.D. thesis.Google Scholar
[14] Segerberg, K., An essay in classical modal logic. Filosofiska Föreningen och Filosofiska Institutienen vid Uppsala Universitet, (1971).Google Scholar
[15] Solovay, R. M., Provability interpretations of modal logic, Israel Journal of Mathematics, vol. 28 (1976), pp. 33–71.Google Scholar