We introduce various kinds of fixed-point properties for modal logics, and we classify the most prominent systems according to these. Our goal is to do a first step towards a complete characterization of provability logics of (possibly non standard) derivability predicates for Peano Arithmetic.

We prove that for i ≥ 1, the arithmetic $I\Delta ₀+{\Omega }_i$ does not prove a variant of its own Herbrand consistency restricted to the terms of depth in $(1+\epsilon )lo{g}^{i+2}$, where ε is an arbitrarily small constant greater than zero. On the other hand, the provability holds for the set of terms of depths in $lo{g}^{i+3}$.