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The polytopologies of transfinite provability logic

(2014) ARCHIVE FOR MATHEMATICAL LOGIC. 53(3-4). p.385-431
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Abstract
Provability logics are modal or polymodal systems designed for modeling the behavior of Godel's provability predicate and its natural extensions. If I > is any ordinal, the Godel-Lob calculus GLP (I >) contains one modality [lambda] for each lambda < I >, representing provability predicates of increasing strength. GLP (omega) has no non-trivial Kripke frames, but it is sound and complete for its topological semantics, as was shown by Icard for the variable-free fragment and more recently by Beklemishev and Gabelaia for the full logic. In this paper we generalize Beklemishev and Gabelaia's result to GLP (I >) for countable I >. We also introduce provability ambiances, which are topological models where valuations of formulas are restricted. With this we show completeness of GLP (I >) for the class of provability ambiances based on Icard polytopologies.
Keywords
Provability logic, Topological semantics, Scattered spaces, Ordinal topology, COMPLETENESS, GLP

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MLA
Fernández-Duque, David. “The Polytopologies of Transfinite Provability Logic.” ARCHIVE FOR MATHEMATICAL LOGIC 53.3-4 (2014): 385–431. Print.
APA
Fernández-Duque, D. (2014). The polytopologies of transfinite provability logic. ARCHIVE FOR MATHEMATICAL LOGIC, 53(3-4), 385–431.
Chicago author-date
Fernández-Duque, David. 2014. “The Polytopologies of Transfinite Provability Logic.” Archive for Mathematical Logic 53 (3-4): 385–431.
Chicago author-date (all authors)
Fernández-Duque, David. 2014. “The Polytopologies of Transfinite Provability Logic.” Archive for Mathematical Logic 53 (3-4): 385–431.
Vancouver
1.
Fernández-Duque D. The polytopologies of transfinite provability logic. ARCHIVE FOR MATHEMATICAL LOGIC. 2014;53(3-4):385–431.
IEEE
[1]
D. Fernández-Duque, “The polytopologies of transfinite provability logic,” ARCHIVE FOR MATHEMATICAL LOGIC, vol. 53, no. 3–4, pp. 385–431, 2014.
@article{8566413,
  abstract     = {Provability logics are modal or polymodal systems designed for modeling the behavior of Godel's provability predicate and its natural extensions. If I > is any ordinal, the Godel-Lob calculus GLP (I >) contains one modality [lambda] for each lambda < I >, representing provability predicates of increasing strength. GLP (omega) has no non-trivial Kripke frames, but it is sound and complete for its topological semantics, as was shown by Icard for the variable-free fragment and more recently by Beklemishev and Gabelaia for the full logic. In this paper we generalize Beklemishev and Gabelaia's result to GLP (I >) for countable I >. We also introduce provability ambiances, which are topological models where valuations of formulas are restricted. With this we show completeness of GLP (I >) for the class of provability ambiances based on Icard polytopologies.},
  author       = {Fernández-Duque, David},
  issn         = {0933-5846},
  journal      = {ARCHIVE FOR MATHEMATICAL LOGIC},
  keywords     = {Provability logic,Topological semantics,Scattered spaces,Ordinal topology,COMPLETENESS,GLP},
  language     = {eng},
  number       = {3-4},
  pages        = {385--431},
  title        = {The polytopologies of transfinite provability logic},
  url          = {http://dx.doi.org/10.1007/s00153-014-0371-1},
  volume       = {53},
  year         = {2014},
}

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