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

We introduce the logics GLP Λ , a generalization of Japaridze's polymodal provability logic GLP ω 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 GLP ω yielding among other things a finitary proof of the normal form theorem for the variablefree fragment of GLP Λ and the decidability of GLP Λ for recursive orderings Λ. Further, we give a restricted axiomatization of the variable-free fragment of GLP Λ .

In this paper we introduce hyperations and cohyperations, which are forms of transfinite iteration of ordinal functions.Hyperations are iterations of normal functions. Unlike iteration by pointwise convergence, hyperation preserves normality. The hyperation f ξ ξ∈On of a normal function f is a sequence of normal functions so that f 0 = id, f 1 = f and for all α, β we have that f α+β = f α f β . These conditions do not determine f α uniquely; in addition, we require that f α α∈On be minimal in an appropriate sense. We study hyperations systematically and show that they are a natural refinement of Veblen progressions.Next, we define cohyperations, very similar to hyperations except that they are left-additive: given α, β, f α+β = f β f α . Cohyperations iterate initial functions which are functions that map initial segments to initial segments. We systematically study co-hyperations and see how they can be employed to define left inverses to hyperations.Hyperations provide an alternative presentation of Veblen progressions and can be useful where a more fine-grained analysis of such sequences is called for. They are very amenable to algebraic manipulation and hence are convenient to work with. Cohyperations, meanwhile, give a novel way to describe slowly increasing functions as often appear, for example, in proof theory.

This paper studies the transfinite propositional provability logics GLPΛ and their corresponding algebras. These logics have for each ordinal ξ < Λ a modality α . We will focus on the closed fragment of GLPΛ (i.e., where no propositional variables occur) and worms therein. Worms are iterated consistency expressions of the form ξn . . . ξ1 ⊤. Beklemishev has defined well-orderings < ξ on worms whose modalities are all at least ξ and presented a calculus to compute the respective order-types.In the current paper we present a generalization of the original < ξ orderings and provide a calculus for the corresponding generalized ordertypes o ξ . Our calculus is based on so-called hyperations which are transfinite iterations of normal functions.Finally, we give two different characterizations of those sequences of ordinals which are of the form o ξ (A) ξ∈On for some worm A. One of these characterizations is in terms of a second kind of transfinite iteration called cohyperation.Definition 2.1. For Λ an ordinal, the logic GLP Λ is the propositional normal modal logic that has for each α < Λ a modality [α] and is axiomatized by all propositional logical tautologies together witht the following schemata: The rules of inference are Modus Ponens and necessitation for each modality:ψ [α]ψ . By GLP we denote the class-size logic that has a modality [α] for each ordinal α and all the corresponding axioms and rules. The classic Gödel-Löb provability logic GL is denoted by GLP 1 . Japaridze algebrasThe relations < α do not give proper linear orders on W α , given that different worms may be equivalent in GLP and hence undistinguishable in the ordering. We remedy this by passing to the Lindenbaum algebra of GLP -that is, the quotient of the language of GLP modulo provable equivalence. This algebra is a Japaridze algebra, as described below:Definition 2.2 (Japaridze algebra). A Japaridze algebra is a structure J = D, {[α]} α<Λ , ∧, ¬, 0, 1 such that

Given a computable ordinal Λ, the transfinite provability logic GLP Λ has for each ξ < Λ a modality [ξ] intended to represent a provability predicate within a chain of increasing strength. One possibility is to read [ξ]φ as φ is provable in T using ω-rules of depth at most ξ, where T is a second-order theory extending ACA 0. In this paper we will formalize such iterations of ω-rules in second-order arithmetic and show how it is a special case of what we call uniform provability predicates. Uniform provability predicates are similar to Ignatiev's strong provability predicates except that they can be iterated transfinitely. Finally, we show that GLP Λ is sound and complete for any uniform provability predicate.