Abstract
The set of all formulas whose n-provability in a given arithmetical theory S is provable in another arithmetical theory T is a recursively enumerable extension of S. We prove that such extensions can be naturally axiomatized in terms of transfinite progressions of iterated local reflection schemata over S. Specifically, the set of all provably 1-provable sentences in Peano arithmetic PA can be axiomatized by an ε0-times iterated local reflection schema over PA. The resulting characterizations provide additional information on the proof-theoretic strength of these theories and on the complexity of their axiomatization.
Similar content being viewed by others
References
L. D. Beklemishev, Theor. Comput. Sci. 224 (1–2) 13–33 (1999).
L. D. Beklemishev, Arch. Math. Logic 42, 515–552 (2003). doi doi 10.1007/s00153-002-0158-7
L. D. Beklemishev, Ann. Pure Appl. Logic 128, 103–123 (2004).
L. D. Beklemishev, Russ. Math. Surv. 60 (2), 197–268 (2005).
L. D. Beklemishev and A. Visser, Ann. Pure Appl. Logic 136 (1–2), 56–74 (2005).
M. Cai, Higher Unprovability (2015).
S. Feferman, J. Symbolic Logic 27, 259–316 (1962).
K. N. Ignatiev, J. Symbolic Logic 58, 249–290 (1993).
G. K. Japaridze, Thesis in Philosophy (Moscow, 1986).
G. Kreisel and A. Lévy, Z. Math. Logik Grundlagen Math. 14, 97–142 (1968).
C. Smorynski, Self-Reference and Modal Logic (Springer-Verlag, Berlin, 1985).
A. M. Turing, Proc. London Math. Soc. Ser. 2 45, 161–228 (1939).
Author information
Authors and Affiliations
Corresponding author
Additional information
Original Russian Text © E.A. Kolmakov, L.D. Beklemishev, 2018, published in Doklady Akademii Nauk, 2018, Vol. 483, No. 3.
Rights and permissions
About this article
Cite this article
Kolmakov, E.A., Beklemishev, L.D. Axiomatizing Provable n-Provability. Dokl. Math. 98, 582–585 (2018). https://doi.org/10.1134/S1064562418070153
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S1064562418070153