For any ordinal $\Lambda$, we can define a polymodal logic $\mathsf{GLP}_\Lambda$, with a modality $[\xi]$ for each $\xi < \Lambda$. These represent provability predicates of increasing strength. Although $\mathsf{GLP}_\Lambda$ has no Kripke models, Ignatiev showed that indeed one can construct a Kripke model of the variable-free fragment with natural number modalities, denoted $\mathsf{GLP}^0_\omega$. Later, Icard defined a topological model for $\mathsf{GLP}^0_\omega$ which is very closely related to Ignatiev's. In this paper we show how to extend these constructions for arbitrary $\Lambda$. (...) More generally, for each $\Theta,\Lambda$ we build a Kripke model $\mathfrak I^\Theta_\Lambda$ and a topological model $\mathfrak T^\Theta_\Lambda$, and show that $\mathsf{GLP}^0_\Lambda$ is sound for both of these structures, as well as complete, provided $\Theta$ is large enough. (shrink)

Ordinal functions may be iterated transfinitely in a natural way by taking pointwise limits at limit stages. However, this has disadvantages, especially when working in the class of normal functions, as pointwise limits do not preserve normality. To this end we present an alternative method to assign to each normal function f a family of normal functions Hyp[f]=〈fξ〉ξ∈OnHyp[f]=〈fξ〉ξ∈On, called its hyperation, in such a way that f0=idf0=id, f1=ff1=f and fα+β=fα∘fβfα+β=fα∘fβ for all α, β.Hyperations are a refinement of the Veblen hierarchy (...) of f. Moreover, if f is normal and has a well-behaved left-inverse g called a left adjoint, then g can be assigned a cohyperationcoH[g]=〈gξ〉ξ∈OncoH[g]=〈gξ〉ξ∈On, which is a family of initial functions such that gξgξ is a left adjoint to fξfξ for all ξ. (shrink)

Provability logics are modal or polymodal systems designed for modeling the behavior of Gödel’s provability predicate and its natural extensions. If Λ is any ordinal, the Gödel-Löb calculus GLPΛ contains one modality [λ] for each λ < Λ, representing provability predicates of increasing strength. GLPω has no non-trivial Kripke frames, but it is sound and complete for its topological semantics, as was shown by Icard for the variable-free fragment and more recently by Beklemishev and Gabelaia for the full logic. In (...) this paper we generalize Beklemishev and Gabelaia’s result to GLPΛ for countable Λ. We also introduce provability ambiances, which are topological models where valuations of formulas are restricted. With this we show completeness of GLPΛ for the class of provability ambiances based on Icard polytopologies. (shrink)

When we interpret modal ◊ as the limit point operator of a topological space, the Gödel-Löb modal system GL defines the class Scat of scattered spaces. We give a partition of Scat into α-slices S α , where α ranges over all ordinals. This provides topological completeness and definability results for extensions of GL. In particular, we axiomatize the modal logic of each ordinal α, thus obtaining a simple proof of the Abashidze–Blass theorem. On the other hand, when we interpret (...) ◊ as closure in a topological space, the Grzegorczyk modal system Grz defines the class HI of hereditarily irresolvable spaces. We also give a partition of HI into α-slices H α , where α ranges over all ordinals. For a subset A of a hereditarily irresolvable space X and an ordinal α, we introduce the α-representation of A, give an axiomatization of the α-representation of A, and characterize H α in terms of α-representations. We prove that ${X \in {\bf H}_{1}}$ iff X is submaximal. For a positive integer n, we generalize the notion of a submaximal space to that of an n-submaximal space, and prove that ${X \in {\bf H}_{n}}$ iff X is n-submaximal. This provides topological completeness and definability results for extensions of Grz. We show that the two partitions are related to each other as follows. For a successor ordinal α = β + n, with β a limit ordinal and n a positive integer, we have ${{\bf H}_{\alpha} \cap {\bf Scat} = {\bf S}_{\beta+2n-1} \cup {\bf S}_{\beta+2n}}$ , and for a limit ordinal α, we have ${{\bf H}_{\alpha} \cap {\bf Scat} = {\bf S}_{\alpha}}$ . As a result, we obtain full and faithful translations of ordinal complete extensions of Grz into ordinal complete extensions of GL, thus generalizing the Kuznetsov–Goldblatt–Boolos theorem. (shrink)