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Models of transfinite provability logic

(2013) JOURNAL OF SYMBOLIC LOGIC. 78(2). p.543-561
Author
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Abstract
For any ordinal Lambda, we can define a polymodal logic GLP(A), with a modality [xi] for each xi < Lambda. These represent provability predicates of increasing strength. Although GLP(A) has no Kripke models, Ignatiev showed that indeed one can construct a Kripke model of the variable-free fragment with natural number modalities, denoted GLP(omega)(0). Later, Icard defined a topological model for GLP(omega)(0) which is very closely related to Ignatiev's. In this paper we show how to extend these constructions for arbitrary Lambda. More generally, for each Theta, Lambda we build a Kripke model J(Lambda)(Theta) and a topological model L-Lambda(Theta), and show that GLP(Lambda)(0) is sound for both of these structures, as well as complete, provided Theta is large enough.
Keywords
CLOSED FRAGMENT

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MLA
Fernández-Duque, David, and Joost J Joosten. “Models of Transfinite Provability Logic.” JOURNAL OF SYMBOLIC LOGIC 78.2 (2013): 543–561. Print.
APA
Fernández-Duque, D., & Joosten, J. J. (2013). Models of transfinite provability logic. JOURNAL OF SYMBOLIC LOGIC, 78(2), 543–561.
Chicago author-date
Fernández-Duque, David, and Joost J Joosten. 2013. “Models of Transfinite Provability Logic.” Journal of Symbolic Logic 78 (2): 543–561.
Chicago author-date (all authors)
Fernández-Duque, David, and Joost J Joosten. 2013. “Models of Transfinite Provability Logic.” Journal of Symbolic Logic 78 (2): 543–561.
Vancouver
1.
Fernández-Duque D, Joosten JJ. Models of transfinite provability logic. JOURNAL OF SYMBOLIC LOGIC. 2013;78(2):543–61.
IEEE
[1]
D. Fernández-Duque and J. J. Joosten, “Models of transfinite provability logic,” JOURNAL OF SYMBOLIC LOGIC, vol. 78, no. 2, pp. 543–561, 2013.
@article{8566420,
  abstract     = {For any ordinal Lambda, we can define a polymodal logic GLP(A), with a modality [xi] for each xi < Lambda. These represent provability predicates of increasing strength. Although GLP(A) has no Kripke models, Ignatiev showed that indeed one can construct a Kripke model of the variable-free fragment with natural number modalities, denoted GLP(omega)(0). Later, Icard defined a topological model for GLP(omega)(0) which is very closely related to Ignatiev's. 
In this paper we show how to extend these constructions for arbitrary Lambda. More generally, for each Theta, Lambda we build a Kripke model J(Lambda)(Theta) and a topological model L-Lambda(Theta), and show that GLP(Lambda)(0) is sound for both of these structures, as well as complete, provided Theta is large enough.},
  author       = {Fernández-Duque, David and Joosten, Joost J},
  issn         = {0022-4812},
  journal      = {JOURNAL OF SYMBOLIC LOGIC},
  keywords     = {CLOSED FRAGMENT},
  language     = {eng},
  number       = {2},
  pages        = {543--561},
  title        = {Models of transfinite provability logic},
  url          = {http://dx.doi.org/10.2178/jsl.7802110},
  volume       = {78},
  year         = {2013},
}

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