Abstract
We investigate what happens when ‘truth’ is replaced with ‘provability’ in Yablo’s paradox. By diagonalization, appropriate sequences of sentences can be constructed. Such sequences contain no sentence decided by the background consistent and sufficiently strong arithmetical theory. If the provability predicate satisfies the derivability conditions, each such sentence is provably equivalent to the consistency statement and to the Gödel sentence. Thus each two such sentences are provably equivalent to each other. The same holds for the arithmetization of the existential Yablo paradox. We also look at a formulation which employs Rosser’s provability predicate.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Beall, J.C. (2001). Is Yablo’s paradox non-circular? Analysis, 61, 176–187.
Cieslinski, C. (2002). Heterologicality and incompleteness. Mathematical Logic Quarterly, 48, 105–110.
Hájek, P., & Pudlák, P. (1998). Metamathematics of first-order arithmetic. Springer.
Ketland, J. (2005). Yablo’s paradox and ω-inconsistency. Synthese, 145(3), 295–302.
Leitgeb, H. (2002). What is a self-referential sentence? Critical remarks on the alleged (non-)circularity of Yablo’s paradox. Logique & Analyse, 177–178, 3–14.
Priest, G. (1997). Yablo’s paradox. Analysis, 57, 236–242.
Smith, P. (2007). An introduction to Gödel’s theorem’s. Cambridge University Press.
Sorensen, R. (1998). Yablo’s paradox and kindred infinite liars. Mind, 107, 137–154.
Urbaniak, R. (2009). Leitgeb, “about”, Yablo. Logique & Analyse, 207, 239–254.
Yablo, S. (1993). Paradox without self–reference. Analysis, 53, 251–252.
Author information
Authors and Affiliations
Corresponding author
Additional information
Graham Leach–Krouse cooperated with us in early stages of this work. Other commitments did not allow him to continue. His short note on the subject is available at http://arxiv.org/abs/1110.2056.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
About this article
Cite this article
Cieśliński, C., Urbaniak, R. Gödelizing the Yablo Sequence. J Philos Logic 42, 679–695 (2013). https://doi.org/10.1007/s10992-012-9244-4
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10992-012-9244-4