Goodstein’s principle is arguably the first purely number-theoretic statement known to be independent of Peano arithmetic. It involves sequences of natural numbers which at first appear to diverge, but eventually decrease to zero. These sequences are defined relative to a notation system based on exponentiation for the natural numbers. In this article, we provide a self-contained and modern analysis of Goodstein’s principle, obtaining some variations and improvements. We explore notions of optimality for notation systems and apply them to the classical Goodstein process and to a weaker variant based on multiplication rather than exponentiation. In particular, we introduce the notion of base-change maximality, and show how it leads to far-reaching extensions of Goodstein’s result. We moreover show that by varying the initial base of the Goodstein process, one readily obtains independence results for each of the fragments $\mathsf {I}\Sigma _n$ of Peano arithmetic.

Modal logic enjoys topological semantics that may be traced back to McKinsey and Tarski, and the classification of topological spaces via modal axioms is a lively area of research. In the past two decades, there has been interest in extending topological modal logic to the language of the mu-calculus, but previously no class of topological spaces was known to be mu-calculus definable that was not already modally definable. In this paper, we show that the full mu-calculus is indeed more expressive than standard modal logic, in the sense that there are classes of topological spaces (and weakly transitive Kripke frames), which are mu-definable but not modally definable. The classes we exhibit satisfy a modally definable property outside of their perfect core, and thus we dub them imperfect spaces. We show that the mu-calculus is sound and complete for these classes. Our examples are minimal in the sense that they use a single instance of a greatest fixed point, and we show that least fixed points alone do not suffice to define any class of spaces that is not already modally definable.

In topological modal logic, it is well known that the Cantor derivative is more expressive than the topological closure, and the ‘elsewhere’, or ‘difference’, operator is more expressive than the ‘somewhere’ operator. In 2014, Kudinov and Shehtman asked whether the combination of closure and elsewhere becomes strictly more expressive when adding the Cantor derivative. In this paper we give an affirmative answer: in fact, the Cantor derivative alone can define properties of topological spaces not expressible with closure and elsewhere. To prove this, we develop a novel theory of morphisms which preserve formulas with the elsewhere operator.

The language of linear temporal logic can be interpreted on the class of dynamic topological systems, giving rise to the intuitionistic temporal logic ${\sf ITL}^{\sf c}_{\Diamond \forall }$ , recently shown to be decidable by Fernández-Duque. In this article we axiomatize this logic, some fragments, and prove completeness for several familiar spaces.

The importance of intuitionistic temporal logics in Computer Science and Artificial Intelligence has become increasingly clear in the last few years. From the proof-theory point of view, intuitionistic temporal logics have made it possible to extend functional programming languages with new features via type theory, while from the semantics perspective, several logics for reasoning about dynamical systems and several semantics for logic programming have their roots in this framework. We consider several axiomatic systems for intuitionistic linear temporal logic and show that each of these systems is sound for a class of structures based either on Kripke frames or on dynamic topological systems. We provide two distinct interpretations of “henceforth”, both of which are natural intuitionistic variants of the classical one. We completely establish the order relation between the semantically defined logics based on both interpretations of “henceforth” and, using our soundness results, show that the axiomatically defined logics enjoy the same order relations.

Let T be a second-order arithmetical theory, Λ a well-order, λ < Λ and X ⊆ ℕ. We use $[\lambda |X]_T^{\rm{\Lambda }}\varphi$ as a formalization of “φ is provable from T and an oracle for the set X, using ω-rules of nesting depth at most λ”.

For a set of formulas Γ, define predicative oracle reflection for T over Γ (Pred–O–RFNΓ(T)) to be the schema that asserts that, if X ⊆ ℕ, Λ is a well-order and φ ∈ Γ, then

$$\forall \,\lambda < {\rm{\Lambda }}\,([\lambda |X]_T^{\rm{\Lambda }}\varphi \to \varphi ).$$

In particular, define predicative oracle consistency (Pred–O–Cons(T)) as Pred–O–RFN{0=1}(T).

Our main result is as follows. Let ATR0 be the second-order theory of Arithmetical Transfinite Recursion, ${\rm{RCA}}_0^{\rm{*}}$ be Weakened Recursive Comprehension and ACA be Arithmetical Comprehension with Full Induction. Then,

$${\rm{ATR}}_0 \equiv {\rm{RCA}}_0^{\rm{*}} + {\rm{Pred - O - Cons\ }}\left( {{\rm{RCA}}_0^{\rm{*}} } \right) \equiv {\rm{RCA}}_0^{\rm{*}} + \,{\rm{Pred - O - Cons\ }}\left( {{\rm{RCA}}_0^{\rm{*}} } \right) \equiv {\rm{RCA}}_0^{\rm{*}} + \,{\rm{Pred - O - RFN}}\,_{{\bf{\Pi }}_2^1 } \left( {{\rm{ACA}}} \right).$$

We may even replace ${\rm{RCA}}_0^{\rm{*}}$ by the weaker ECA0, the second-order analogue of Elementary Arithmetic.

Thus we characterize ATR0, a theory often considered to embody Predicative Reductionism, in terms of strong reflection and consistency principles.

Given a scattered space $\mathfrak{X} = \left( {X,\tau } \right)$ and an ordinal λ, we define a topology $\tau _{ + \lambda } $ in such a way that τ +0 = τ and, when $\mathfrak{X}$ is an ordinal with the initial segment topology, the resulting sequence {τ +λ}λ∈Ord coincides with the family of topologies $\left\{ {\mathcal{I}_\lambda } \right\}_{\lambda \in Ord} $ used by Icard, Joosten, and the second author to provide semantics for polymodal provability logics.

We prove that given any scattered space $\mathfrak{X}$ of large-enough rank and any ordinal λ > 0, GL is strongly complete for τ + λ. The special case where $\mathfrak{X} = \omega ^\omega + 1$ and λ = 1 yields a strengthening of a theorem of Abashidze and Blass.

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 . Later, Icard defined a topological model for 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 and a topological model , and show that is sound for both of these structures, as well as complete, provided Θ is large enough.

Dynamic Topological Logic () is a multimodal system for reasoning about dynamical systems. It is defined semantically and, as such, most of the work done in the field has been model-theoretic. In particular, the problem of finding a complete axiomatization for the full language of over the class of all dynamical systems has proven to be quite elusive.

Here we propose to enrich the language to include a polyadic topological modality, originally introduced by Dawar and Otto in a different context. We then provide a sound axiomatization for over this extended language, and prove that it is complete. The polyadic modality is used in an essential way in our proof.

Dynamic Topological Logic is a modal framework for reasoning about dynamical systems, that is, pairs 〈X, f〉 where X is a topological space and f: X → X a continuous function.

In this paper we consider the case where X is a metric space. We first show that any formula which can be satisfied on an arbitrary dynamic topological system can be satisfied on one based on a metric space; in fact, this space can be taken to be countable and have no isolated points. Since any metric space with these properties is homeomorphic to the set of rational numbers, it follows that any satisfiable formula can be satisfied on a system based on ℚ.

We then show that the situation changes when considering complete metric spaces, by exhibiting a formula which is not valid in general but is valid on the class of systems based on a complete metric space. While we do not attempt to give a full characterization of the set of valid formulas on this class we do give a relative completeness result; any formula which is satisfiable on a dynamical system based on a complete metric space is also satisfied on one based on the Cantor space.