For every natural number n we introduce a new weak choice principle $\mathrm {nRC_{fin}}$:

Given any infinite set x, there is an infinite subset $y\subseteq x$ and a selection function f that chooses an n-element subset from every finite $z\subseteq y$ containing at least n elements.

By constructing new permutation models built on a set of atoms obtained as Fraïssé limits, we will study the relation of $\mathrm {nRC_{fin}}$ to the weak choice principles $\mathrm {RC_m}$ (that has already been studied in [3] and [6]):

Given any infinite set x, there is an infinite subset $y\subseteq x$ with a choice function f on the family of all m-element subsets of y.

Moreover, we prove a stronger analogue of the results in [6] when we study the relation between $\mathrm {nRC_{fin}}$ and $\mathrm {kC_{fin}^-}$ which is defined by

Given any infinite family $\mathcal {F}$ of finite sets of cardinality greater than k, there is an infinite subfamily $\mathcal {A}\subseteq \mathcal {F}$ with a selection function f that chooses a k-element subset from each $A\in \mathcal {A}$.

We construct a model in which all C-sequences are trivial, yet there exists a $\kappa $-Souslin tree all of whose limit levels are vanishing levels. This provides an optimal combination of compactness and incompactness. It is obtained by incorporating a so-called mutually exclusive ascent path to Kunen’s 1978 forcing construction, and by analyzing a gallery of $\kappa $-cc forcing extensions of the outcome model.

Let X be an uncountable Polish space and let $\mathcal {I}$ be an ideal on $\omega $. A point $\eta \in X$ is an $\mathcal {I}$-limit point of a sequence $(x_n)$ taking values in X if there exists a subsequence $(x_{k_n})$ convergent to $\eta $ such that the set of indexes $\{k_n: n \in \omega \}\notin \mathcal {I}$. Denote by $\mathscr {L}(\mathcal {I})$ the family of subsets $S\subseteq X$ such that S is the set of $\mathcal {I}$-limit points of some sequence taking values in X or S is empty. In this article, we study the relationships between the topological complexity of ideals $\mathcal {I}$, their combinatorial properties, and the families of sets $\mathscr {L}(\mathcal {I})$ which can be attained. On the positive side, we provide several purely combinatorial (not depending on the space X) characterizations of ideals $\mathcal {I}$ for the inclusions and the equalities between $\mathscr {L}(\mathcal {I})$ and the Borel classes $\Pi ^0_1$, $\Sigma ^0_2$, and $\Pi ^0_3$. As a consequence, we prove that if $\mathcal {I}$ is a $\Pi ^0_4$ ideal then exactly one of the following cases holds: $\mathscr {L}(\mathcal {I})=\Pi ^0_1$ or $\mathscr {L}(\mathcal {I})=\Sigma ^0_2$ or $\mathscr {L}(\mathcal {I})=\Sigma ^1_1$ (however we do not have an example of a $\Pi ^0_4$ ideal with $\mathscr {L}(\mathcal {I})=\Sigma ^1_1$). In addition, we provide an explicit example of a coanalytic ideal $\mathcal {I}$ for which $\mathscr {L}(\mathcal {I})=\Sigma ^1_1$. On the negative side, since $\mathscr {L}(\mathcal {I})$ contains all singletons, it is immediate that there are no ideals $\mathcal {I}$ such that $\mathscr {L}(\mathcal {I})=\Sigma ^0_1$. On the same direction, we show that there are no ideals $\mathcal {I}$ such that $\mathscr {L}(\mathcal {I})=\Pi ^0_2$ or $\mathscr {L}(\mathcal {I})=\Sigma ^0_3$. In fact, for instance, if $\mathcal {I}$ is a Borel ideal and $\mathscr {L}(\mathcal {I})$ contains a non $\Sigma ^0_2$ set, then it contains all $\Pi ^0_3$ sets. We conclude with several open questions.

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].

We extend the framework of abstract algebraic logic to weak logics, namely, logical systems that are not necessarily closed under uniform substitution. We interpret weak logics by algebras expanded with an additional predicate, and we introduce a loose and strict version of algebraizability for weak logics. We study this framework by investigating the connection between the algebraizability of a weak logic and the algebraizability of its schematic fragment, and we then prove a version of Blok and Pigozzi’s Isomorphism Theorem in our setting. We apply this framework to logics in team semantics and show that the classical versions of inquisitive and dependence logic are strictly algebraizable, while their intuitionistic versions are only loosely so.

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$_2$.

Let $ \lambda $ be an uncountable cardinal such that $ 2^{< \lambda } = \lambda $. Working in the setup of generalized descriptive set theory, we study the structure of $ \lambda ^+ $-Borel measurable functions with respect to various kinds of limits, and isolate a suitable notion of $ \lambda $-Baire class $ \xi $ function. Among other results, we provide higher analogues of two classical theorems of Lebesgue, Hausdorff, and Banach, namely:

1. (1) A function is $ \lambda ^+ $-Borel measurable if and only if it can be obtained from continuous functions by iteratively applying pointwise $ D $-limits, where $ D $ varies among directed sets of size at most $ \lambda $.

2. (2) A function is of $ \lambda $-Baire class $ \xi $ if and only if it is $ \lambda ^+\text {-}\boldsymbol {\Sigma }^{ 0 }_{ \xi +1} $-measurable.

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.