We study the computational problem of rigorously describing the asymptotic behavior of topological dynamical systems up to a finite but arbitrarily small pre-specified error. More precisely, we consider the limit set of a typical orbit, both as a spatial object (attractor set) and as a statistical distribution (physical measure), and we prove upper bounds on the computational resources of computing descriptions of these objects with arbitrary accuracy. We also study how these bounds are affected by different dynamical constraints and provide several examples showing that our bounds are sharp in general. In particular, we exhibit a computable interval map having a unique transitive attractor with Cantor set structure supporting a unique physical measure such that both the attractor and the measure are non-computable.

Some informal arguments are valid, others are invalid. A core application of logic is to tell us which is which by capturing these validity facts. Philosophers and logicians have explored how well a host of logics carry out this role, familiar examples being propositional, first-order and second-order logic. Since natural language and standard logics are countable, a natural question arises: is there a countable logic guaranteed to capture the validity patterns of any language fragment? That is, is there a countable omega-universal logic? Our article philosophically motivates this question, makes it precise, and then answers it. It is a self-contained, concise sequel to ‘Capturing Consequence’ by A.C. Paseau (RSL vol. 12, 2019).

Using iterated Sacks forcing and topological games, we prove that the existence of a totally imperfect Menger set in the Cantor cube with cardinality continuum is independent from ZFC. We also analyze the structure of Hurewicz and consonant subsets of the Cantor cube in the Sacks model.

The purpose of this paper is to investigate forcing as a tool to construct universal models. In particular, we look at theories of initial segments of the universe and show that any model of a sufficiently rich fragment of those theories can be embedded into a model constructed by forcing. Our results rely on the model-theoretic properties of good ultrafilters, for which we provide a new existence proof on non-necessarily complete Boolean algebras.

This paper focuses on the structurally complete extensions of the system $\mathbf {R}$-mingle ($\mathbf {RM}$). The main theorem demonstrates that the set of all hereditarily structurally complete extensions of $\mathbf {RM}$ is countably infinite and forms an almost-chain, with only one branching element. As a corollary, we show that the set of structurally complete extensions of $\mathbf {RM}$ that are not hereditary is also countably infinite and forms a chain. Using algebraic methods, we provide a complete description of both sets. Furthermore, we offer a characterization of passive structural completeness among the extensions of $\mathbf {RM}$: specifically, we prove that a quasivariety of Sugihara algebras is passively structurally complete if and only if it excludes two specific algebras. As a corollary, we give an additional characterization of quasivarieties of Sugihara algebras that are passively structurally complete but not structurally complete. We close the paper with a characterization of actively structurally complete quasivarieties of Sugihara algebras.

We investigate the end extendibility of models of arithmetic with restricted elementarity. By utilizing the restricted ultrapower construction in the second-order context, for each $n\in \mathbb {N}$ and any countable model of $\mathrm {B}\Sigma _{n+2}$, we construct a proper $\Sigma _{n+2}$-elementary end extension satisfying $\mathrm {B}\Sigma _{n+1}$, which answers a question by Clote positively. We also give a characterization of the countable models of $\mathrm {I}\Sigma _{n+2}$ in terms of their end extendibility, similar to the case of $\mathrm {B}\Sigma _{n+2}$. Along the proof, we introduce a new type of regularity principle in arithmetic called the weak regularity principle, which serves as a bridge between the model’s end extendibility and the amount of induction or collection it satisfies.

We investigate Steel’s conjecture in ‘The Core Model IterabilityProblem’ [10], that if $\mathcal {W}$ and $\mathcal {R}$ are $\Omega +1$-iterable, $1$-small weasels, then $\mathcal {W}\leq ^{*}\mathcal {R}$ iff there is a club $C\subset \Omega $ such that for all $\alpha \in C$, if $\alpha $ is regular, then $\alpha ^{+\mathcal {W}}\leq \alpha ^{+\mathcal {R}}$. We will show that the conjecture fails, assuming that thereis an iterable premouse M which models KP and which has a-Woodin cardinal. On the other hand, we show thatassuming there is no transitive model of KP with a Woodin cardinal theconjecture holds. In the course of this we will also show that ifM is a premouse which models KP with a largest, regular,uncountable cardinal $\delta $, and $\mathbb {P} \in M$ is a forcing poset such that $M\models "\mathbb {P}\text { has the }\delta \text {-c.c.}"$, and $g\subset \mathbb {P}$ is M-generic, then $M[g]\models \text {KP}$. Additionally, we study the preservation of admissibilityunder iteration maps. At last, we will prove a fact about the closure of the setof ordinals at which a weasel has the S-hull property. Thisanswers another question implicit in remarks in [10].

We show that if a field A is not pseudo-finite, then there is no prime model of the theory of pseudo-finite fields over A. Assuming GCH, we extend this result to $\kappa $-prime models, for $\kappa $ an uncountable cardinal or $\aleph _\varepsilon $.

We argue that logicism, the thesis that mathematics is reducible to logic and analytic truths, is true. We do so by (a) developing a formal framework with comprehension and abstraction principles, (b) giving reasons for thinking that this framework is part of logic, (c) showing how the denotations for predicates and individual terms of an arbitrary mathematical theory can be viewed as logical objects that exist in the framework, and (d) showing how each theorem of a mathematical theory can be given an analytically true reading in the logical framework.

We argue that some of Brouwer’s assumptions, rejected by Bishop, should be considered and studied as possible axioms. We show that Brouwer’s Continuity Principle enables one to prove an intuitionistic Borel Hierarchy Theorem. We also explain that Brouwer’s Fan Theorem is useful for a development of the theory of measure and integral different from the one worked out by Bishop. We show that Brouwer’s bar theorem not only proves the Fan Theorem but also a stronger statement that we call the Almost-fan Theorem. The Almost-fan Theorem implies intuitionistic versions of Ramsey’s Theorem and the Bolzano-Weierstrass Theorem.

We study H-structures associated with $SU$-rank 1 measurable structures. We prove that the $SU$-rank of the expansion is continuous and that it is uniformly definable in terms of the parameters of the formulas. We also introduce notions of dimension and measure for definable sets in the expansion and prove they are uniformly definable in terms of the parameters of the formulas.

We prove various iteration theorems for forcing classes related to subproper and subcomplete forcing, introduced by Jensen. In the first part, we use revised countable support iterations, and show that 1) the class of subproper, ${}^\omega \omega $-bounding forcing notions, 2) the class of subproper, T-preserving forcing notions (where T is a fixed Souslin tree) and 3) the class of subproper, $[T]$-preserving forcing notions (where T is an $\omega _1$-tree) are iterable with revised countable support. In the second part, we adopt Miyamoto’s theory of nice iterations, rather than revised countable support. We show that this approach allows us to drop a technical condition in the definitions of subcompleteness and subproperness, still resulting in forcing classes that are iterable in this way, preserve $\omega _1$, and, in the case of subcompleteness, don’t add reals. Further, we show that the analogs of the iteration theorems proved in the first part for RCS iterations hold for nice iterations as well.

We say that a Kripke model is a GL-model (Gödel and Löb model) if the accessibility relation $\prec $ is transitive and converse well-founded. We say that a Kripke model is a D-model if it is obtained by attaching infinitely many worlds $t_1, t_2, \ldots $, and $t_\omega $ to a world $t_0$ of a GL-model so that $t_0 \succ t_1 \succ t_2 \succ \cdots \succ t_\omega $. A non-normal modal logic $\mathbf {D}$, which was studied by Beklemishev [3], is characterized as follows. A formula $\varphi $ is a theorem of $\mathbf {D}$ if and only if $\varphi $ is true at $t_\omega $ in any D-model. $\mathbf {D}$ is an intermediate logic between the provability logics $\mathbf {GL}$ and $\mathbf {S}$. A Hilbert-style proof system for $\mathbf {D}$ is known, but there has been no sequent calculus. In this paper, we establish two sequent calculi for $\mathbf {D}$, and show the cut-elimination theorem. We also introduce new Hilbert-style systems for $\mathbf {D}$ by interpreting the sequent calculi. Moreover, we show that D-models can be defined using an arbitrary limit ordinal as well as $\omega $. Finally, we show a general result as follows. Let X and $X^+$ be arbitrary modal logics. If the relationship between semantics of X and semantics of $X^+$ is equal to that of $\mathbf {GL}$ and $\mathbf {D}$, then $X^+$ can be axiomatized based on X in the same way as the new axiomatization of $\mathbf {D}$ based on $\mathbf {GL}$.

The class of all $\ast $-continuous Kleene algebras, whose description includes an infinitary condition on the iteration operator, plays an important role in computer science. The complexity of reasoning in such algebras—ranging from the equational theory to the Horn one, with restricted fragments of the latter in between—was analyzed by Kozen (2002). This paper deals with similar problems for $\ast $-continuous residuated Kleene lattices, also called $\ast $-continuous action lattices, where the product operation is augmented by residuals. We prove that, in the presence of residuals, the fragment of the corresponding Horn theory with $\ast $-free hypotheses has the same complexity as the $\omega ^\omega $ iteration of the halting problem, and hence is properly hyperarithmetical. We also prove that if only commutativity conditions are allowed as hypotheses, then the complexity drops down to $\Pi ^0_1$ (i.e., the complement of the halting problem), which is the same as that for $\ast $-continuous Kleene algebras. In fact, we get stronger upper bound results: the fragments under consideration are translated into suitable fragments of infinitary action logic with exponentiation, and our upper bounds are obtained for the latter ones.

This paper provides a consistent first-order theory solving the knower paradoxes of Kaplan and Montague, with the following main features: 1. It solves the knower paradoxes by providing a faithful formalization of the principle of veracity (that knowledge requires truth), using both a knowledge and a truth predicate. 2. It is genuinely untyped i.e., it is untyped not only in the sense that it uses a single knowledge predicate applying to all sentences in the language (including sentences in which this predicate occurs), but in the sense that its axioms quantify over all sentences in the language, thus supporting comprehensive reasoning with untyped knowledge ascriptions. 3. Common knowledge predicates can be defined in the system using self-reference. These facts, together with a technique based on Löb’s theorem, enables it to support comprehensive reasoning with untyped common knowledge ascriptions (without having any axiom directly addressing common knowledge).

In this work, we investigate various combinatorial properties of Borel ideals on countable sets. We extend a theorem presented in [13] and identify an $F_\sigma $ tall ideal in which player II has a winning strategy in the Cut and Choose Game, thereby addressing a question posed by J. Zapletal. Additionally, we explore the Ramsey properties of ideals, demonstrating that the random graph ideal is critical for the Ramsey property when considering more than two colors. The previously known result for two colors is extended to any finite number of colors. Furthermore, we comment on the Solecki ideal and identify an $F_\sigma $ tall K-uniform ideal that is not equivalent to $\mathcal {ED}_{\text {fin}}$, thereby addressing a question from M. Hrušák’s work [10].

In this paper, we show that the existence of certain first-countable compact-like extensions is equivalent to the equality between corresponding cardinal characteristics of the continuum. For instance, $\mathfrak b=\mathfrak s=\mathfrak c$ if and only if every regular first-countable space of weight $< \mathfrak c$ can be densely embedded into a regular first-countable countably compact space.

Let $\operatorname {TFAb}_r$ be the class of torsion-free abelian groups of rank r, and let $\operatorname {FD}_r$ be the class of fields of characteristic $0$ and transcendence degree r. We compare these classes using various notions. Considering the Scott complexity of the structures in the classes and the complexity of the isomorphism relations on the classes, the classes seem very similar. Hjorth and Thomas showed that the $\operatorname {TFAb}_r$ are strictly increasing under Borel reducibility. This is not so for the classes $\operatorname {FD}_r$. Thomas and Velickovic showed that for sufficiently large r, the classes $\operatorname {FD}_r$ are equivalent under Borel reducibility. We try to compare the groups with the fields, using Borel reducibility, and also using some effective variants. We give functorial Turing computable embeddings of $\operatorname {TFAb}_r$ in $\operatorname {FD}_r$, and of $\operatorname {FD}_r$ in $\operatorname {FD}_{r+1}$. We show that under computable countable reducibility, $\operatorname {TFAb}_1$ lies on top among the classes we are considering. In fact, under computable countable reducibility, isomorphism on $\operatorname {TFAb}_1$ lies on top among equivalence relations that are effective $\Sigma _3$.

This paper studies which truth-values are most likely to be taken on finite models by arbitrary sentences of a many-valued predicate logic. The classical zero-one law (independently proved by Fagin and Glebskiĭ et al.) states that every sentence in a purely relational language is almost surely false or almost surely true, meaning that the probability that the formula is true in a randomly chosen finite structures of cardinal n is asymptotically $0$ or $1$ as n grows to infinity. We obtain generalizations of this result for any logic with values in a finite lattice-ordered algebra, and for some infinitely valued logics, including Łukasiewicz logic. The finitely valued case is reduced to the classical one through a uniform translation and Oberschelp’s generalization of the zero-one law. Moreover, it is shown that the complexity of determining the almost sure value of a given sentence is PSPACE-complete (generalizing Grandjean’s result for the classical case), and for some logics we describe completely the set of truth-values that can be taken by sentences almost surely.

We study versions of the tree pigeonhole principle, $\mathsf {TT}^1$, in the context of Weihrauch-style computable analysis. The principle has previously been the subject of extensive research in reverse mathematics, an outstanding question of which investigation is whether $\mathsf {TT}^1$ is $\Pi ^1_1$-conservative over the ordinary pigeonhole principle, $\mathsf {RT}^1$. Using the recently introduced notion of the first-order part of an instance-solution problem, we formulate the analog of this question for Weihrauch reducibility, and give an affirmative answer. In combination with other results, we use this to show that unlike $\mathsf {RT}^1$, the problem $\mathsf {TT}^1$ is not Weihrauch requivalent to any first-order problem. Our proofs develop new combinatorial machinery for constructing and understanding solutions to instances of $\mathsf {TT}^1$.