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.

We introduce the notion of a weak A2 space (or wA2-space), which generalises spaces satisfying Todorčević’s axioms A1–A4 and countable vector spaces. We show that in any Polish weak A2 space, analytic sets are Kastanas Ramsey, and discuss the relationship between Kastanas Ramsey sets and sets in the projective hierarchy. We also show that in all spaces satisfying A1–A4, every subset of $\mathcal {R}$ is Kastanas Ramsey iff Ramsey, generalising the recent result by [2]. Finally, we show that in the setting of Gowers wA2-spaces, Kastanas Ramsey sets and strategically Ramsey sets coincide, providing a connection between the recent studies on topological Ramsey spaces and countable vector spaces.

In this note, we prove that Kim-dividing over models is always witnessed by a coheir Morley sequence, whenever the theory is NATP.

Following the strategy of Chernikov and Kaplan [8], we obtain some corollaries which hold in NATP theories. Namely, (i) if a formula Kim-forks over a model, then it quasi-divides over the same model and (ii) for any tuple of parameters b and a model M, there exists a global coheir p containing $\text {tp}(b/M)$ such that for all $b'\models p|_{MB}$.

We also show that for coheirs in NATP theories, condition (ii) above is a necessary condition for being a witness of Kim-dividing, assuming that a witness of Kim-dividing exists (see Definition 4.1 in this note). That is, if we assume that a witness of Kim-dividing always exists over any given model, then a coheir $p\supseteq \text {tp}(a/M)$ must satisfy (ii) whenever it is a witness of Kim-dividing of a over a model M. We also give a sufficient condition for the existence of a witness of Kim-dividing in terms of pre-independence relations.

At the end of the article, we leave a short remark on Mutchnik’s recent work [17]. We point out that the class of N-$\omega $-DCTP$_2$ theories, a subclass of the class of NATP theories, contains all NTP$_2$ theories and NSOP$_1$ theories. We also note that Kim-forking and Kim-dividing are equivalent over models in N-$\omega $-NDCTP$_2$ theories, where Kim-dividing is defined with respect to invariant Morley sequences, instead of coheir Morley sequences as in [17].

This article is concerned with finite rank stability theory, and more precisely two classical ways to decompose a type using minimal types. The first is its domination equivalence to a Morley product of minimal types, and the second is its semi-minimal analysis, both of which are useful in applications. Our main interest is to explore how these two decompositions are connected. We prove that neither determine the other in general, and give more precise connections using various notions from the model theory literature such as uniform internality, proper fibrations, and disintegratedness.

We consider the problem of predicting the next bit in an infinite binary sequence sampled from the Cantor space with an unknown computable measure. We propose a new theoretical framework to investigate the properties of good computable predictions, focusing on such predictions’ convergence rate.

Since no computable prediction can be the best, we first define a better prediction as one that dominates the other measure. We then prove that this is equivalent to the condition that the sum of the KL divergence errors of its predictions is smaller than that of the other prediction for more computable measures. We call that such a computable prediction is more general than the other.

We further show that the sum of any sufficiently general prediction errors is a finite left-c.e. Martin-Löf random real. This means the errors converge to zero more slowly than any computable function.

Within the determinacy setting, ${\mathscr {P}({\omega _1})}$ is regular (in the sense of cofinality) with respect to many known cardinalities and thus there is substantial evidence to support the conjecture that ${\mathscr {P}({\omega _1})}$ has globally regular cardinality. However, there is no known information about the regularity of ${\mathscr {P}(\omega _2)}$. It is not known if ${\mathscr {P}(\omega _2)}$ is even $2$-regular under any determinacy assumptions. The article will provide the following evidence that ${\mathscr {P}(\omega _2)}$ may possibly be ${\omega _1}$-regular: Assume $\mathsf {AD}^+$. If $\langle A_\alpha : \alpha < {\omega _1} \rangle $ is such that ${\mathscr {P}(\omega _2)} = \bigcup _{\alpha < {\omega _1}} A_\alpha $, then there is an $\alpha < {\omega _1}$ so that $\neg (|A_\alpha | \leq |[\omega _2]^{<\omega _2}|)$.

Building on the correspondence between finitely axiomatised theories in Łukasiewicz logic and rational polyhedra, we prove that the unification type of the fragment of Łukasiewicz logic with $n\geqslant 2$ variables is nullary. This solves a problem left open by V. Marra and L. Spada [Ann. Pure Appl. Logic 164 (2013), pp. 192–210]. Furthermore, we refine the study of unification with bounds on the number of variables. Our proposal distinguishes the number m of variables allowed in the problem and the number n in the solution. We prove that the unification type of Łukasiewicz logic for all $m,n \geqslant 2$ is nullary.

This article is a contribution to the “neostability” type of result for abstract elementary classes. Under certain set theoretic assumptions, we propose a definition and a characterization of NIP in AECs. The class of AECs with NIP properly contains the class of stable AECs.1 We show that for an AEC K and $\lambda \geq LS(K)$, $K_\lambda $ is NIP if and only if there is a notion of nonforking on it which we call a w*-good frame. On the other hand, the negation of NIP leads to being able to encode subsets.

A longstanding question is to characterize the lattice of supersets (modulo finite sets), $\mathcal {L}^*(A)$, of a low$_2$ computably enumerable (c.e.) set. The conjecture is that $\mathcal {L}^*(A)\cong {\mathcal E}^*$. In spite of claims in the literature, this longstanding question/conjecture remains open. We contribute to this problem by solving one of the main test cases. We show that if c.e. A is low$_2$ then A has an atomless hyperhypersimple superset. In fact, if A is c.e. and low$_2$, then for any $\Sigma _3$-Boolean algebra B there is some c.e. $H\supseteq A$ such that $\mathcal {L}^*(H)\cong B$.

We show that the class of Krasner hyperfields is not elementary. To show this, we determine the rational rank of quotients of multiplicative groups in field extensions. We also discuss some related questions.

For any $2 \le n < \omega $, we introduce a forcing poset using generalized promises which adds a normal n-splitting subtree to a $(\ge \! n)$-splitting normal Aronszajn tree. Using this forcing poset, we prove several consistency results concerning finitely splitting subtrees of Aronszajn trees. For example, it is consistent that there exists an infinitely splitting Suslin tree whose topological square is not Lindelöf, which solves an open problem due to Marun. For any $2 < n < \omega $, it is consistent that every $(\ge \! n)$-splitting normal Aronszajn tree contains a normal n-splitting subtree, but there exists a normal infinitely splitting Aronszajn tree which contains no $(< \! n)$-splitting subtree. To show the latter consistency result, we prove a forcing iteration preservation theorem related to not adding new small-splitting subtrees of Aronszajn trees.

Gödel’s completeness theorem for classical first-order logic is one of the most basic theorems of logic. Central to any foundational course in logic, it connects the notion of valid formula, i.e., a formula satisfied in all models, to the notion of provable formula.

We survey a few standard formulations and proofs of the completeness theorem before focusing on the formal description of a slight modification of Henkin’s proof within intuitionistic second-order arithmetic.

It is usual, in the context of the completeness of intuitionistic logic with respect to various semantics, such as Kripke or Beth semantics, to follow the Curry–Howard correspondence and to interpret the proofs of completeness as programs which turn proofs of validity for these semantics into proofs of derivability.

We apply this approach to Henkin’s proof to phrase it as a program which transforms any proof of validity with respect to Tarski semantics into a proof of derivability.

By doing so, we hope to shed some “effective” light on the relation between Tarski semantics and syntax: proofs of validity are syntactic objects with which we can compute.

We investigate the notion of ideal (equivalently: filter) Schauder basis of a Banach space. We do so by providing bunch of new examples of such bases that are not the standard ones, especially within classical Banach spaces ($\ell _p$, $c_0$, and James’ space). Those examples lead to distinguishing and characterizing ideals (equivalently: filters) in terms of Schauder bases. We investigate the relationship between possibly basic sequences and ideals (equivalently: filters) on the set of natural numbers.

Working within the context of countable, superstable theories, we give many equivalents of a theory having NOTOP. In particular, NOTOP is equivalent to V-DI, the assertion that any type V-dominated by an independent triple is isolated over the triple. If T has NOTOP, then every model N is atomic over an independent tree of countable, elementary substructures, and hence is determined up to back-and-forth equivalence over such a tree. We also verify Shelah’s assertion from Chapter XII of [9] that NOTOP implies PMOP (without using NDOP).

For cardinals $\mathfrak {a}$ and $\mathfrak {b}$, we write $\mathfrak {a}=^\ast \mathfrak {b}$ if there are sets A and B of cardinalities $\mathfrak {a}$ and $\mathfrak {b}$, respectively, such that there are partial surjections from A onto B and from B onto A. $=^\ast $-equivalence classes are called surjective cardinals. In this article, we show that $\mathsf {ZF}+\mathsf {DC}_\kappa $, where $\kappa $ is a fixed aleph, cannot prove that surjective cardinals form a cardinal algebra, which gives a negative solution to a question proposed by Truss [J. Truss, Ann. Pure Appl. Logic 27, 165–207 (1984)]. Nevertheless, we show that surjective cardinals form a “surjective cardinal algebra”, whose postulates are almost the same as those of a cardinal algebra, except that the refinement postulate is replaced by the finite refinement postulate. This yields a smoother proof of the cancellation law for surjective cardinals, which states that $m\cdot \mathfrak {a}=^\ast m\cdot \mathfrak {b}$ implies $\mathfrak {a}=^\ast \mathfrak {b}$ for all cardinals $\mathfrak {a},\mathfrak {b}$ and all nonzero natural numbers m.

We investigate some variants of the splitting, reaping, and independence numbers defined using asymptotic density. Specifically, we give a proof of Con($\mathfrak {i}<\mathfrak {s}_{1/2}$), Con($\mathfrak {r}_{1/2}<\mathfrak {b}$), and Con($\mathfrak {i}_*<2^{\aleph _0}$). This answers two questions raised in [5]. Besides, we prove the consistency of $\mathfrak {s}_{1/2}^{\infty } < $ non$(\mathcal {E})$ and cov$(\mathcal {E}) < \mathfrak {r}_{1/2}^{\infty }$, where $\mathcal {E}$ is the $\sigma $-ideal generated by closed sets of measure zero.

We prove that the structure $(\mathbb {Z},<,+,R)$ is distal for all congruence-periodic sparse predicates $R\subseteq \mathbb {N}$. We do so by constructing a strong honest definition for every formula $\phi (x;y)$ with $\lvert {x}\rvert =1$, providing a rare example of concrete distal decompositions.

In this work, we consider the ideals $m^0(\mathcal {I})$ and $\ell ^0(\mathcal {I})$, ideals generated by the $\mathcal {I}$-positive Miller trees and $\mathcal {I}$-positive Laver trees, respectively. We investigate in which cases these ideals have cofinality larger than $\mathfrak {c}$ and we calculate some cardinal invariants closely related to these ideals.