Abstract
Fixing some computably enumerable theory \(T\), the Friedman-Goldfarb-Harrington (FGH) theorem says that over elementary arithmetic, each \({\varSigma }_1\) formula is equivalent to some formula of the form \(\Box _T \varphi \) provided that \(T\) is consistent. In this paper we give various generalizations of the FGH theorem. In particular, for \(n>1\) we relate \({\varSigma }_{n}\) formulas to provability statements \([n]^{\mathsf{True}}_T\varphi \) which are a formalization of “provable in \(T\) together with all true \({\varSigma }_{n+1}\) sentences”. As a corollary we conclude that each \([n]^{\mathsf{True}}_T\) is \({\varSigma }_{n+1}\)-complete. This observation yields us to consider a recursively defined hierarchy of provability predicates \([n+1]^\Box _T\) which look a lot like \([n+1]^{\mathsf{True}}_T\) except that where \([n+1]^{\mathsf{True}}_T\) calls upon the oracle of all true \({\varSigma }_{n+2}\) sentences, the \([n+1]^\Box _T\) recursively calls upon the oracle of all true sentences of the form \(\langle n \rangle _T^\Box \phi \). As such we obtain a ‘syntax-light’ characterization of \({\varSigma }_{n+1}\) definability whence of Turing jumps which is readily extended beyond the finite. Moreover, we observe that the corresponding provability predicates \([n+1]_T^\Box \) are well behaved in that together they provide a sound interpretation of the polymodal provability logic \({\mathsf {GLP}} _\omega \).
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References
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Acknowledgements
I would like to thank Lev Beklemishev, Ramon Jansana, Stephen Simpson and Albert Visser for encouragement and fruitful discussions. También quisiera agradecerles a Diego Agulló Castelló y Rosa María Espinosa Jaén, alcaldes de las pedanías de Elche, Maitino y Perleta respectivamente, por facilitarme un sitio donde trabajar durante el verano del 2014. The research was supported by the Generalitat de Catalunya under grant number 2014SGR437 and from the Spanish Ministry of Science and Education under grant numbers MTM2011-26840, and MTM2011-25747.
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Joosten, J.J. (2015). Turing Jumps Through Provability. In: Beckmann, A., Mitrana, V., Soskova, M. (eds) Evolving Computability. CiE 2015. Lecture Notes in Computer Science(), vol 9136. Springer, Cham. https://doi.org/10.1007/978-3-319-20028-6_22
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DOI: https://doi.org/10.1007/978-3-319-20028-6_22
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