In previous work [4] we introduced and examined the class of betweenness algebras. In the current article we study a larger class of algebras with binary operators of possibility and sufficiency, the weak mixed algebras. Furthermore, we develop a system of logic with two binary modalities, sound and complete with respect to the class of frames closely related to the aforementioned algebras, and we prove an embedding theorem which solves an open problem from [4].

In [3], Hjorth proved from $ZF + AD + DC$ that there is no sequence of distinct $\boldsymbol {\Sigma ^1_2}$ sets of length $\boldsymbol {\delta ^1_2}$. Sargsyan [11] extends Hjorth’s technique to show there is no sequence of distinct $\boldsymbol {\Sigma ^1_{2n}}$ sets of length $\boldsymbol {\delta ^1_{2n}}$. Sargsyan conjectured an analogous property is true for any regular Suslin pointclass in $L(\mathbb {R})$—i.e., if $\kappa $ is a regular Suslin cardinal in $L(\mathbb {R})$, then there is no sequence of distinct $\kappa $-Suslin sets of length $\kappa ^+$ in $L(\mathbb {R})$. We prove this in the case that the pointclass $S(\kappa )$ is inductive-like.

Partition regularity over algebraic structures is a topic in Ramsey theory that has been extensively researched by combinatorialists [2, 3, 5, 15]. Motivated by recent work in this area, we investigate the computability-theoretic and reverse-mathematical aspects of partition regularity over algebraic structures—an area that, to the best of our knowledge, has not been explored before. This article focuses on a 1975 theorem by Straus [25], which has played a significant role in many of the results in this field.

The motivation of this article is to introduce a kind of orbit equivalence relations which can well describe structures and properties of Polish groups from the perspective of Borel reducibility. Given a Polish group G, let $E(G)$ be the right coset equivalence relation $G^\omega /c(G)$, where $c(G)$ is the group of all convergent sequences in G. Let G be a Polish group. (1) G is a discrete countable group containing at least two elements iff $E(G)\sim _BE_0$; (2) if G is TSI uncountable non-Archimedean, then $E(G)\sim _BE_0^\omega $; (3) G is non-Archimedean iff $E(G)\le _B=^+$; (4) if H is a CLI Polish group but G is not, then $E(G)\not \le _BE(H)$; and (5) if H is a non-Archimedean Polish group but G is not, then $E(G)\not \le _BE(H)$. The notion of $\alpha $-l.m.-unbalanced Polish group for $\alpha <\omega _1$ is introduced. Let G and H be Polish groups, $0<\alpha <\omega _1$. If G is $\alpha $-l.m.-unbalanced but H is not, then $E(G)\not \le _B E(H)$. For TSI Polish groups, the existence of Borel reduction is transformed into the existence of a well-behaved continuous mapping between topological groups. As its applications, for any Polish group G, let $G_0$ be the connected component of the identity element $1_G$. Let G and H be two separable TSI Lie groups. If $E(G)\le _BE(H)$, then there exists a continuous locally injective map $S:G_0\to H_0$. Moreover, if $G_0$ and $H_0$ are abelian, S is a group homomorphism. In particular, for $c_0,e_0,c_1,e_1\in {\mathbb {N}}$, $E({\mathbb {R}}^{c_0}\times {\mathbb {T}}^{e_0})\le _BE({\mathbb {R}}^{c_1}\times {\mathbb {T}}^{e_1})$ iff $e_0\le e_1$ and $c_0+e_0\le c_1+e_1$.

We show that the statement “In every separable pseudometric space there is a maximal non-strictly $\delta $-separated set.” implies the axiom of choice for countable families of sets. This gives answers to a question of Dybowski and Górka [2]. We also prove several related results.

Kurt Gödel proved that it is not possible to characterize intuitionistic propositional logic (${IPL}$) by means of finite and deterministic truth-tables. After extending the same result with respect to non-deterministic matrices (Nmatrices), we provide a semantical characterization of ${IPL}$ by means of a $3$-valued Nmatrix with a restricted set of valuations. This structure allows to define an algorithm to delete unsound rows from the non-deterministic truth-tables generated for each formula, which constitutes a new and very simple decision procedure for ${IPL}$. This method can be seen as truth-tables in a broader sense, and a way to overcome Gödel’s limiting result.

We provide a partial answer to a question asked independently by Kim and d’Elbée and show that, under the assumption of the stable Kim-forking conjecture, every $\text {NSOP}_1$ rosy theory must be simple. We also prove that the theory of a Frobenius field has stable Kim-forking.

Goldstern showed in [7] that the union of a real-parametrized, monotone family of Lebesgue measure zero sets has also Lebesgue measure zero, provided that the sets are uniformly $\boldsymbol {\Sigma }^1_1$. Our aim is to study to what extent we can drop the $\boldsymbol {\Sigma }^1_1$ assumption. We show that Goldstern’s principle for the pointclass $\boldsymbol {\Pi }^1_1$ holds. We show that Goldstern’s principle for the pointclass of all subsets is consistent with $\mathsf {ZFC}$ and show its negation follows from $\mathsf {CH}$. Also we prove that Goldstern’s principle for the pointclass of all subsets holds both under $\mathsf {ZF} + \mathsf {AD}$ and in Solovay models.

In this article we study the theories of the infinite-branching tree and the r-regular tree, and show that both of them are pseudofinite. Moreover, we show that they can be realized by infinite ultraproducts of polynomial exact classes of graphs, and provide a characterization of the Morley rank of definable sets in terms of the degrees of polynomials measuring their non-standard cardinalities. This answers negatively some questions from [2], where it is asked whether every stable generalised measurable structure is one-based.

We prove a common refinement of theorems of Bergfalk and of Casarosa and Lambie-Hanson, showing that under certain hypotheses, the higher derived limits of a certain inverse system of abelian groups $\mathbf {A}$ do not vanish. The refined theorem has a number of interesting corollaries, including the nonvanishing of the second derived limit of $\mathbf {A}$ in many of the common models of set theory of the reals and in the Mitchell model. In particular, we disprove a conjecture of Bergfalk, Hrušák, and Lambie-Hanson that higher derived limits of $\mathbf {A}$ vanish in the Miller model.

Suppose $G\curvearrowright X$ is a Polish group action, H is a Polish group, and $G\times X\mathop {\overset {\psi }\longrightarrow } H$ is a cocycle that is continuous in the second variable. If $\psi $ is either Baire measurable or is $\lambda \times \mu $-measurable with respect to a Haar measure $\lambda $ on G and a fully supported $\sigma $-finite Borel measure $\mu $ on X, then $\psi $ is jointly continuous.

This article studies the question of which classes of numberings can be generated by the direct sums of computable uniformly minimal sequences of numberings (in particular, Friedberg and positive numberings). For the class of Friedberg numberings, this question was initiated by Britta Schinzel in her 1982 paper. In this article, we show that the class of Gödel numberings is generated by the direct sums of computable uniformly positive sequences of universal numberings and that there exists a conull class of oracles computing sequences of Friedberg numberings with programmable direct sums. We further show that all computable numberings of a fairly wide class of families of total recursive functions (containing, for example, the family of all primitive recursion functions) are generated by the direct sums of computable sequences of their incomparable Friedberg numberings. On the other hand we prove that no family of partial recursive functions has a computable sequence of Friedberg numberings whose direct sum is acceptable.

This article deals with the class of existentially closed models of fields with a distinguished submodule (over a fixed subring). In the positive characteristic case, this class is elementary and was investigated by the first-named author. Here we study this class in Robinson’s logic, meaning the category of existentially closed models with embeddings following Haykazyan and Kirby, and prove that in this context this class is NSOP$_1$ and TP$_2$.

Except for a limited number of cases, a complete classification of the Diophantine (i.e., positive existentially definable) sets of polynomial rings and fields of rational functions seems out of reach at present. We contribute to this problem by proving that several natural sets and relations over these structures are not Diophantine. For instance, we show that the relation of equality of degrees is not Diophantine over the field of complex rational functions in one variable and, in the same structure, we show that certain family of relations that approximates the valuation ring at infinity is not Diophantine either.

We extend the logical categories framework to first-order modal logic. In our modal categories, modal operators are applied directly to subobjects and interact with the background factorization system. We prove a Joyal-style representation theorem into relational structures formalizing a ‘counterpart’ notion. We investigate saturation conditions related to definability questions and we enrich our framework with quotients and disjoint sums, thus leading to the notion of a modal (quasi) pretopos. We finally show a way to build syntactic categories out of first-order modal theories.

We explore the Weihrauch degree of the problems “find a bad sequence in a non-well quasi order” ($\mathsf {BS}$) and “find a descending sequence in an ill-founded linear order” ($\mathsf {DS}$). We prove that $\mathsf {DS}$ is strictly Weihrauch reducible to $\mathsf {BS}$, correcting our mistaken claim in [18]. This is done by separating their respective first-order parts. On the other hand, we show that $\mathsf {BS}$ and $\mathsf {DS}$ have the same finitary and deterministic parts, confirming that $\mathsf {BS}$ and $\mathsf {DS}$ have very similar uniform computational strength. We prove that König’s lemma $\mathsf {KL}$ and the problem $\mathsf {wList}_{{2^{\mathbb {N}}},\leq \omega }$ of enumerating a given non-empty countable closed subset of ${2^{\mathbb {N}}}$ are not Weihrauch reducible to $\mathsf {DS}$ or $\mathsf {BS}$, resolving two main open questions raised in [18]. We also answer the question, raised in [12], on the existence of a “parallel quotient” operator, and study the behavior of $\mathsf {BS}$ and $\mathsf {DS}$ under the quotient with some known problems.

By a celebrated result of Kučera and Slaman [5], the Martin-Löf random left-c.e. reals form the highest left-c.e. Solovay degree. Barmpalias and Lewis-Pye [1] strengthened this result by showing that, for all left-c.e. reals $\alpha $ and $\beta $ such that $\beta $ is Martin-Löf random and all left-c.e. approximations $a_0,a_1,\dots $ and $b_0,b_1,\dots $ of $\alpha $ and $\beta $, respectively, the limit

$$ \begin{align*} \lim\limits_{n\to\infty}\frac{\alpha - a_n}{\beta - b_n} \end{align*} $$

$\alpha $

$\beta $

Here we give an equivalent formulation of the result of Barmpalias and Lewis-Pye in terms of nondecreasing translation functions and generalize their result to the set of all (i.e., not necessarily left-c.e.) reals.

Normal modal logics extending the logic $\mathsf {K4.3}$ of linear transitive frames are known to lack the Craig interpolation property (CIP), except some logics of bounded depth such as $\mathsf {S5}$. We turn this ‘negative’ fact into a research question and pursue a non-uniform approach to Craig interpolation by investigating the following interpolant existence problem: decide whether there exists a Craig interpolant between two given formulas in any fixed logic above $\mathsf {K4.3}$. Using a bisimulation-based characterisation of interpolant existence for descriptive frames, we show that this problem is decidable and coNP-complete for all finitely axiomatisable normal modal logics containing $\mathsf {K4.3}$. It is thus not harder than entailment in these logics, which is in sharp contrast to other recent non-uniform interpolation results. We also extend our approach to Priorean temporal logics (with both past and future modalities) over the standard time flows—the integers, rationals, reals, and finite strict linear orders—none of which is blessed with the CIP.