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Well-orders in the transfinite Japaridze algebra

(2014) LOGIC JOURNAL OF THE IGPL. 22(6). p.933-963
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Abstract
This article studies the transfinite propositional provability logics GLP(A) and their corresponding algebras. These logics have for each ordinal xi < Lambda a modality <xi >. We will focus on the closed fragment of GLP(Lambda) (i.e. where no propositional variables occur) and worms therein. Worms are iterated consistency expressions of the form <xi(n)>...<xi(1)> inverted perpendicular. Beklemishev has defined well-orderings < (xi) on worms whose modalities are all at least xi and presented a calculus to compute the respective order-types. In the current article, we present a generalization of the original <(xi) orderings and provide a calculus for the corresponding generalized order-types o(xi). Our calculus is based on so-called hyperations which are transfinite iterations of normal functions. Finally, we give two different characterizations of those sequences of ordinals which are of the form < 0(xi) (A)>(xi epsilon On) for some worm A. One of these characterizations is in terms of a second kind of transfinite iteration called cohyperation.
Keywords
Provability logics, ordinal notations, Japaridze algebra, reflection calculus, PROVABILITY LOGIC

Citation

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MLA
Fernández-Duque, David, and Joost J Joosten. “Well-orders in the Transfinite Japaridze Algebra.” LOGIC JOURNAL OF THE IGPL 22.6 (2014): 933–963. Print.
APA
Fernández-Duque, D., & Joosten, J. J. (2014). Well-orders in the transfinite Japaridze algebra. LOGIC JOURNAL OF THE IGPL, 22(6), 933–963.
Chicago author-date
Fernández-Duque, David, and Joost J Joosten. 2014. “Well-orders in the Transfinite Japaridze Algebra.” Logic Journal of the Igpl 22 (6): 933–963.
Chicago author-date (all authors)
Fernández-Duque, David, and Joost J Joosten. 2014. “Well-orders in the Transfinite Japaridze Algebra.” Logic Journal of the Igpl 22 (6): 933–963.
Vancouver
1.
Fernández-Duque D, Joosten JJ. Well-orders in the transfinite Japaridze algebra. LOGIC JOURNAL OF THE IGPL. 2014;22(6):933–63.
IEEE
[1]
D. Fernández-Duque and J. J. Joosten, “Well-orders in the transfinite Japaridze algebra,” LOGIC JOURNAL OF THE IGPL, vol. 22, no. 6, pp. 933–963, 2014.
@article{8566415,
  abstract     = {This article studies the transfinite propositional provability logics GLP(A) and their corresponding algebras. These logics have for each ordinal xi < Lambda a modality <xi >. We will focus on the closed fragment of GLP(Lambda) (i.e. where no propositional variables occur) and worms therein. Worms are iterated consistency expressions of the form <xi(n)>...<xi(1)> inverted perpendicular. Beklemishev has defined well-orderings < (xi) on worms whose modalities are all at least xi and presented a calculus to compute the respective order-types. 
In the current article, we present a generalization of the original <(xi) orderings and provide a calculus for the corresponding generalized order-types o(xi). Our calculus is based on so-called hyperations which are transfinite iterations of normal functions. 
Finally, we give two different characterizations of those sequences of ordinals which are of the form < 0(xi) (A)>(xi epsilon On) for some worm A. One of these characterizations is in terms of a second kind of transfinite iteration called cohyperation.},
  author       = {Fernández-Duque, David and Joosten, Joost J},
  issn         = {1367-0751},
  journal      = {LOGIC JOURNAL OF THE IGPL},
  keywords     = {Provability logics,ordinal notations,Japaridze algebra,reflection calculus,PROVABILITY LOGIC},
  language     = {eng},
  number       = {6},
  pages        = {933--963},
  title        = {Well-orders in the transfinite Japaridze algebra},
  url          = {http://dx.doi.org/10.1093/jigpal/jzu018},
  volume       = {22},
  year         = {2014},
}

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