Turing Jumps Through Provability

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 \({\mathsf {GLP}} _\omega \).