Papers

Sequential prediction naturally induces an innovation sequence consisting of the prediction errors produced by a causal predictor. We use the collection of all such innovation sequences to define the innovation spectrum of an individual binary sequence and, from it, a new reducibility based on sequential information extraction. We show that this reducibility refines truth-table reducibility while exhibiting a fundamentally different geometry. The degree structure decomposes into two canonical regions: a truth-table spine, whose induced order is isomorphic to the truth-table degrees, and a complementary reservoir-immune region, consisting of sequences from which no infinite computable predictable reservoir can be extracted. We establish bridge constructions connecting the two regions, prove that reservoir immunity is preserved under sequential innovation, and show that the Martin–Löf-random degrees form a proper downward-closed substructure inside the reservoir-immune region. These results reveal a new geometric organization of individual sequences based on causal predictability rather than classical oracle computation.

Partial orthomodular lattices are an intermediate class of algebras between ortholattices and orthomodular lattices. In this short note, we obtain a duality for partial orthomodular lattices via a subcategory of the category of spectral spaces, independently of the Axiom of Choice.

Goodstein's theorem states that a certain sequences based on exponential notation for the natural numbers are always finite. The result is independent of Peano arithmetic and is a prototypical example of a proof of termination by transfinite induction. A variant based instead on the Ackermann function has more recently been proposed by Arai et al., and instead is independent of the more powerful theory ${\sf ATR}_0$. However, this result is contingent on rather elaborate normal forms for natural numbers based on a `sandwiching' procedure. This leaves open both the question of whether the sandwiching procedure can be eliminated while retaining the full strength of the Ackermannian Goodstein principle, and whether other normal forms can lead to non-termination. In this article we settle these questions by showing that any Goodstein process based on the Ackermann function is terminating, and indeed the sandwiching procedure gives rise to Goodstein principles of maximal length. We thus obtain an equivalent principle which does not involve normal forms at all and immediately implies all Ackermannian Goodstein principles that have been considered. Our techniques provide a new approach to termination proofs, where terms in a sequence do not necessarily decrease in complexity, but instead are majorized by some “master” process, already known to be terminating.

We investigate the encoding ability of the cohesive power construction. We compute a graph $\mathcal{G}$ where the cohesive power $\prod_C \mathcal{G}$ of $\mathcal{G}$ by any $\Delta_2$ cohesive set $C$ has degree $0''$. That is, $0''$ computes a presentation of $\prod_C \mathcal{G}$, and every presentation of $\prod_C \mathcal{G}$ computes $0''$. We also compute a linear order $\mathcal{L}$ where no cohesive power of $\mathcal{L}$ has a computable presentation. We accomplish this by ensuring that if $\mathcal{P}$ is a presentation of a cohesive power of $\mathcal{L}$, then $\mathcal{P}''$ has $\mathrm{PA}$-degree relative to $0''$.

Just as Visser showed that the formal propositional logic $\mathbf{FPL}$ can be embedded into Gödel-Löb provability logic $\mathbf{GL}$, Petrukhin proposed a propositional logic $\mathbf{SPL}$ that can be embedded into Solovay's non-normal provability logic $\mathbf{S}$. In this paper, we fix Petrukhin's proof and extend the result to Japaridze's provability logic $\mathbf{D}$, and propose a propositional logic $\mathbf{DPL}$ that can be embedded into $\mathbf{D}$.

This paper deals with the model theory of convolution algebras $(L^1(G),*)$ for locally compact groups $G$, seen as Banach lattices equipped with the convolution product $*$. We first prove transfer principles for elementary equivalence and elementary embeddings when the underlying group $G$ is discrete, namely, $(\ell^1(G),*) \equiv (\ell^1(H),*)$ implies $G \equiv H$, while the converse holds when $G$ and $H$ are $\omega$-saturated (likewise for elementary substructures). Without $\omega$-saturation, the converse fails. Although pure Banach lattices are model-theoretically tame, our results imply that adding convolution yields wild behavior. For example, we prove that if $G$ is any locally compact, non-discrete group, then the formula $d(x,x*y)$ is unstable with respect to $\mathrm{Th}(L^1(G),*)$. Moreover, we show that if $G$ is discrete and contains a particular configuration of amenable subgroups, then the formula $d(x*y,z)\mathbin{\dot{-}}\frac{1}{2}$ witnesses $\mathrm{TP}_2$ with respect to $\mathrm{Th}(\ell^1(G),*)$. As a consequence, if $G$ contains an infinite abelian subgroup, then $\mathrm{Th}(\ell^{1}(G),*)$ has $\mathrm{TP}_{2}$. We prove similar results in the locally compact non-discrete setting using the notion of an approximate identity. Finally, we prove a `continuous-by-discrete' approximation theorem. Namely, convolution algebras of connected abelian Lie groups admit metric embeddings into ultraproducts of convolution algebras over finite abelian groups.

Gödel's ontological argument is modal by design. Possibility and necessity are among its central concerns, and connected interpretation decisions – extensions or intensions for the positive properties, constant or varying quantifier domains, the choice of modal logic, etc. – subtly shape the argument and, in particular, whether the modal collapse is implied or not. It is tempting to abstract the modality away and to read the positive properties as merely forming an ultrafilter – which, however, may replace Gödel's argument with a perhaps more manageable alternative that may have little to do with Gödel's original intention. In this spirit, Odifreddi and Gomes argue that modal collapse is an intrinsic structural feature of any theory that characterises positive properties as an ultrafilter and identifies God as its principal generator. This comment shows, with machine-verified counterexamples, that the structural claim is false: what drives the collapse is primarily not the filter structure but the rigidity of positivity – itself a thoroughly modal condition. Moreover, two claims of Odifreddi and Gomes are corrected: one on the derivability of Gödel's Theorem IV, one on the extensionality of positivity. All results are machine-checked in Isabelle/HOL.

In this paper, we study when uniform interpretations and definable quotients preserve finite-dimensional asymptotic classes. Counting in the ambient structures gives finitely many asymptotic alternatives. However, the corresponding parameter sets need not be definable in the interpreted language, so in general the interpreted class is only weakly asymptotic. We define trace reflection, a one-sided condition which ensures this definability. We first prove a transfer theorem for interpretations with definable selectors and give explicit bounds on the denominator. We then use invariant profiles to prove a transfer theorem for general definable quotients, without assuming definable choice. We define an intrinsic visible denominator and prove that it is the least asymptotic denominator. We also give a coprime-witness criterion for the interpretation-dependent bound to be sharp. We prove formula-wise, syntactic, and semantic criteria for descent, as well as composition theorems, and show that trace reflection is strictly weaker than uniform weak bi-interpretability. Using profiles of finite frames, we remove the noncanonical parameters used to reconstruct finite-field coordinates. As applications, we study projective quotient traces and show that, for every fixed $k\geq2$, the pure incidence structures given by graphs of polynomials of degree less than $k$ over finite fields form a full $k$-dimensional asymptotic class. The denominator $k$ is minimal, and every member is $K_{k,n}$-free for all $n\geq2$.

Let $K$ be any quadratic number field, and let $O_K$ be the ring of algebraic integers in $K$. In 1975 J. Denef proved that Hilbert's Tenth Problem over $O_K$ has a negative solution. In this paper we establish the following undecidability result: There is no algorithm to decide whether an arbitrarily given polynomial equation $P(z_1,\ldots,z_{16})=0$ (with integer coefficients and 16 unknowns) has solutions over $O_K$. Moreover, when $K$ is a real quadratic field, we show that $15$ unknowns suffice for undecidability.

We organize and generalize several dualities involving continuous posets. The main theorem reads $\mathbf{St}_\alpha\mathbf{Inf}_{\alpha'}\mathbf{Cont}_{\beta'}\mathbf{Sup}_\beta \simeq (\mathbf{St}_\beta\mathbf{Inf}_{\beta'}\mathbf{Cont}_{\alpha'}\mathbf{Sup}_\alpha)^{\mathrm{op}}$, where $\mathbf{St}$, $\mathbf{Inf}$, $\mathbf{Cont}$ and $\mathbf{Sup}$ refer to stability, completeness, continuity and cocompleteness. The indices are "ladders", i.e., classes of sets $\lambda$ stable under dependent sums and quotients, with associated notions of $\lambda$-small infima and ${\lambda}$-filtered suprema. In the second half of the paper, we discuss algebraicity, proximity lattices and perfect maps.

The purpose of this study is to investigate the bitopological duality theory of De Morgan algebras equipped with a closure operator, known as S4 De Morgan algebras. We first introduce certain expansions of pairwise Stone spaces, which we call pairwise S4 De Morgan Stone spaces (henceforth, PS4D-spaces). These consist of a pairwise Stone space $X$ equipped with a twist continuous involution $g\colon X\to X$, as well as a binary relation $R\subseteq X\times X$ that is reflexive and transitive. We first demonstrate that the bitopological spectrum $S_0(A)$ of prime filters of an S4 De Morgan algebra $A$ gives rise to a PS4D-space. A topological representation is then obtained by exhibiting an isomorphism from $A$ to the S4 De Morgan algebra $A_0(S_0(A))$ of $(\tau_1,\delta_2)$-biclopen subsets of $S_0(A)$ whose operation of De Morgan involution is defined through $g$ and whose closure operator is defined through $R$. We then provide an algebraic realization theorem by showing that every PS4D-space $X$ is bihomeomorphic and relationally isomorphic to the bitopological spectrum $S_0(A_0(X))$ of prime filters of $A_0(X)$. With the introduction of suitable bicontinuous frame morphisms, we show that the category $\mathbf{S4D}$ of S4 De Morgan algebras is dually equivalent to the category $\mathbf{PStone_{S4D}}$ of PS4D-spaces. As an application, we provide bitopological characterizations of filters and ideals in general De Morgan algebras under our established duality as well as bitopological soundness and completeness results for an S4-type modal extension of the calculus FDE of first-degree entailment.

We study a refinement of the analytic hierarchy over second-order arithmetic under the arithmetical Church thesis, the assertion that every set of natural numbers is arithmetical. The thesis is false in the full standard model, but it is naturally satisfied in the $\omega$-model consisting of the arithmetical sets. Over $\mathsf{ACA}_0^\ast+\mathsf{ACT}$, second-order quantifiers can be replaced by quantification over codes for arithmetical sets, and this gives normal forms which distinguish first-order and second-order quantifier alternations more finely than the usual analytic hierarchy. As an application, we use these normal forms to answer a question of Frittaion on fragments of uniform reflection in second-order arithmetic: the choice assumption in his separation theorem cannot simply be omitted.

For each countable ordinal $\alpha\geq 2$, Filipów, Kowalczuk and Kwela introduced an ideal $\mathsf{conv}_\alpha$ on the countable compact ordinal space $\omega^\alpha+1$. Kowalczuk later proved that, for each countable limit ordinal $\lambda$, the ideal $\mathsf{conv}_{<\lambda}$ is the greatest lower bound of $\{\mathsf{conv}_\beta:\beta<\lambda\}$ in the Katětov order. At the first uncountable level, let $A\subseteq[2,\omega_1)$ be uncountable and let $D$ be a countable dense subset of $X_A=\prod_{\alpha\in A}(\omega^\alpha+1)$. The coordinate ideal $\mathsf{Conv}(A,D)$ on $D$ consists of those $B\subseteq D$ with $\pi_\alpha[B]\in\mathsf{conv}_\alpha$ for every $\alpha\in A$. For a pair $D\subseteq E$ of countable dense sets, call $\alpha$ non-small if $\pi_\alpha[E\setminus D]\notin\mathsf{conv}_\alpha$. In ZFC, if at most countably many coordinates are non-small, then $\mathsf{Conv}(A,D)\equiv_K\mathsf{Conv}(A,E)$. Under CH this countability bound is sharp: for every $A\subseteq[3,\omega_1)$ with $|A|=\aleph_1$, there are countable dense sets $D\subseteq D^*\subseteq X_A$ such that $\mathsf{Conv}(A,D^*)\leq_K\mathsf{Conv}(A,D)$ but $\mathsf{Conv}(A,D)\not\leq_K\mathsf{Conv}(A,D^*)$, and in particular $\mathsf{Conv}(A,D)$ and $\mathsf{Conv}(A,D^*)$ are not Katětov equivalent. The non-reduction is obtained, under CH, by diagonalizing along $\omega_1$ coordinates against the elements of $\omega^\omega$ that code retractions $D^*\to D$.

A real number can have more than one name: $0.5$ and $0.4999\ldots$ denote the same thing, and which numbers enjoy such a doubling depends on the base one writes in. We make that dependence the object of study. Fixing a system function $\vartheta$, which assigns a base $\vartheta(n)\geq2$ to every position independently, we build the real numbers as equivalence classes of digit functions $f:\omega\rightarrow\omega$, where the equivalence is generated by two local carrying moves, contraction and broadening, and tested by agreement on finite initial segments. Each class turns out to contain at most two canonical representatives (up to a sign representation), its primary and secondary auxiliary functions, so the doubling above is a theorem of the theory rather than a convention imposed on it. We define order, addition and multiplication and verify the axioms of a Dedekind-complete ordered field, so that by categoricity every choice of $\vartheta$ delivers $\mathbb{R}$ itself. The models are therefore indistinguishable as ordered fields and differ only in how their elements are named, and since $\vartheta$ ranges over an uncountable parameter space, that naming can be chosen to suit a problem. We show what this buys: in any base whose partial products absorb every denominator, a non-zero real number is rational precisely when it has a second name, and Cantor's 1869 irrationality criterion for Cantor series follows from the representation theory rather than from number theory.

We introduce an ordinal-valued invariant of well-founded trees, called their type. We use type to give an alternative proof that the class of all well-founded trees is well-quasi-ordered by the homomorphism relation, and we establish a relationship between type and the usual ordinal rank of a well-founded tree, showing in particular that type provides a strictly coarser structural classification.

In recent years, there has been rapid development in the foundational study of oracle computability from the perspective of Lawvere-Tierney topologies and their sheaves. In this article, we formulate and analyze the notion of reducibility within the framework of total computability. Then, using sheaf subtoposes derived from oracles in the total computable setting, we establish separations between various hierarchies of logical principles, including the hierarchies of the weak law of excluded middle $\mathbf{WLEM}$, the lessor limited principle of omniscience $\mathbf{LLPO}$, and Markov's principle $\mathbf{MP}$.

We develop a cell decomposition framework for o-minimal structures equipped with a generic derivation. To a $\delta$-cell we associate source cells and finite configurations in ordinary o-minimal sorts, allowing differential-topological questions to be studied through finite jet spaces. We then introduce a metric space of definable curve germs and identify its local half-space pieces with Cartesian powers of the maximal ideal of the Hardy field of definable germs. Using this germ-space description, we prove an abstract curve selection theorem for the $\delta$-topology. In the case of closed ordered differential fields, we further describe concrete asymptotic representatives for the abstract curve germs.