On Provability Logics with Linearly Ordered Modalities

We introduce the logics GLPΛ, a generalization of Japaridze’s polymodal provability logic GLPω where Λ is any linearly ordered set representing a hierarchy of provability operators of increasing strength. We shall provide a reduction of these logics to GLPω yielding among other things a finitary proof of the normal form theorem for the variable-free fragment of GLPΛ and the decidability of GLPΛ for recursive orderings Λ. Further, we give a restricted axiomatization of the variable-free fragment of GLPΛ.