We study Martsinkovsky–Russell torsion modules with pure embeddings as an abstract elementary class. We give a model-theoretic characterization of the pure-injective and the Σ-pure-injective modules relative to the class of torsion modules assuming that the torsion submodule is a pure submodule. Our characterization of relative Σ-pure-injective modules extends the classical characterization of Gruson and Jenson as well as Zimmermann.

We study the limit models of the class and determine when the class is superstable assuming that the torsion submodule is a pure submodule. As a corollary, we show that the class of torsion abelian groups with pure embeddings is strictly stable (i.e., stable not superstable).

This paper belongs to the research on the limit of the first incompleteness theorem. Effectively inseparable ($\mathsf{EI}$) theories can be viewed as an effective version of essentially undecidable ($\mathsf{EU}$) theories, and $\mathsf{EI}$ is stronger than $\mathsf{EU}$. We examine this question: Are there minimal effectively inseparable theories with respect to interpretability? We propose $\mathsf{tEI}$, the theory version of $\mathsf{EI}$. We first prove that there are no minimal $\mathsf{tEI}$ theories with respect to interpretability (i.e., for any $\mathsf{tEI}$ theory T, we can effectively find a theory which is $\mathsf{tEI}$ and strictly weaker than T with respect to interpretability). By a theorem due to Pour-EI, we have that $\mathsf{tEI}$ is equivalent with $\mathsf{EI}$. Thus, there are no minimal $\mathsf{EI}$ theories with respect to interpretability. Also, we prove that there are no minimal finitely axiomatizable $\mathsf{EI}$ theories with respect to interpretability.

We study the preservation of certain properties under products of classes of finite structures. In particular, we examine indivisibility, definable self-similarity, the amalgamation property, and the disjoint n-amalgamation property. We explore how each of these properties interacts with the lexicographic product, full product, and free superposition of classes of structures. Additionally, we consider the classes of theories which admit configurations indexed by these products. In particular, we show that, under mild assumptions, the products considered in this article do not yield new classes of theories.

Like many intensional logics, William Parry’s logic of analytic implication $\mathsf{PAI}$ admits extensions determined by imposing semantic conditions on its account of modality. $\mathsf{PAI}$ is unique, however, in its allowing a second dimension—a topic-theoretic dimension—along which extensions can be defined. The recent introduction by Francesco Berto of topic-sensitive intentional modals (TSIMs)—which disagree with $\mathsf{PAI}$ on this type of condition—provide further motivations to examine such topic-theoretic extensions. In this paper, we introduce, motivate, and characterize a number of such extensions of $\mathsf{PAI}$, paying special attention to applications to Berto’s framework and to ways in which the modal and topic-theoretic dimensions influence one another.

In this article, I first analyze and assess the epistemological and semantic status of canonical value-range equations in the formal language of Frege’s Grundgesetze der Arithmetik. I subsequently scrutinize the relation between (a) his informal, metalinguistic stipulation in Grundgesetze I, Section 3, and (b) its formal counterpart, which is Basic Law V. One point I argue for is that the stipulation in Section 3 was designed not only to fix the references of value-range names, but that it was probably also intended to play a regulative role with an eye to Basic Law V. I further intend to shed new light on the status of Frege’s identification of the truth-values with their unit classes in Grundgesetze I, Section 10 and determine its place and importance in his overall logicist project. In a subsequent section, I first discuss the hypothetical extendability of the logical system of Grundgesetze. In what follows, I analyze the incompleteness of that system—were it not inconsistent and hence trivially complete—by focusing on the role of the definite description operator and the axiom governing it (= Basic Law VI) as well as on the role of the application operator. The latter is defined by means of the former. I further comment on the redundancy of an axiom that could be designed to supplement Basic Law VI by incorporating the second clause of Frege’s two-part elucidation of the description operator. I argue that although it seems likely that Basic Law VI could have been shown to be dispensable in Grundgesetze, Frege would probably have been reluctant to dispense with it, despite his commitment to axiomatic parsimony. In my view, the reason is that he considers it essential that every primitive function-name of his formal language be governed and grounded by a basic law that is tailored to the nature and the role of the primitive name occurring at a key point in the law. Toward the end of the article, I propose solutions to two problems with Frege’s use of “=” in the concept-script sentences that result from his definitions involving the replacement of the double stroke of definition with the judgment-stroke. I conclude the article with a summary of the main results that I have achieved.

In this article, I develop a novel version of the multiverse theory of sets called hierarchical pluralism by introducing the notion of “degrees of intentionality” of theories. The presented view is articulated for the purpose of reconciling epistemological realism and the multiverse theory of sets so as to preserve a considerable amount of epistemic objectivity when working with the multiverse theory. I give some arguments in favor of a hierarchical picture of the multiverse in which theories or models are thought to be ordered with respect to their plausibility as a manifestation of endorsing the idea that some set theories are more plausible than others. The proposed multiverse account settles the pluralist’s dilemma, the dichotomy that there is a trade-off between the richness of mathematical ontology and the objectivity of mathematical truth. The view also extends and serves as an alternative position to Balaguer’s intention-based Platonism, from which he claims that a certain version of mathematical pluralism follows.

We consider interpretable topological spaces and topological groups in a p-adically closed field K. We identify a special class of “admissible topologies” with topological tameness properties like generic continuity, similar to the topology on definable subsets of $K^n$. We show that every interpretable set has at least one admissible topology, and that every interpretable group has a unique admissible group topology. We then consider definable compactness (in the sense of Fornasiero) on interpretable groups. We show that an interpretable group is definably compact if and only if it has finitely satisfiable generics (fsg), generalizing an earlier result on definable groups. As a consequence, we see that fsg is a definable property in definable families of interpretable groups, and that any fsg interpretable group defined over ${\mathbb{Q}}_p$ is definably isomorphic to a definable group.