- Author
- David Fernández-Duque (UGent) and Joost J Joosten
- Organization
- 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
Please use this url to cite or link to this publication: http://hdl.handle.net/1854/LU-8566415
- 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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