  Fixing some computably enumerable theory $T$, the
Friedman-Goldfarb-Harrington (FGH) theorem says that over elementary
arithmetic, each $\Sigma_1$ 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
$\Sigma_{n}$ formulas to provability statements $[n]_T^{\sf True}\varphi$ which
are a formalization of "provable in $T$ together with all true $\Sigma_{n+1}$
sentences". As a corollary we conclude that each $[n]_T^{\sf True}$ is
$\Sigma_{n+1}$-complete. This observation yields us to consider a recursively
defined hierarchy of provability predicates $[n+1]^\Box_T$ which look a lot
like $[n+1]_T^{\sf True}$ except that where $[n+1]_T^{\sf True}$ calls upon the
oracle of all true $\Sigma_{n+2}$ 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
$\Sigma_{n+1}$ 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 ${\sf GLP}_\omega$.
