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$_2$.

This article studies relative unification and admissibility in intuitionistic logic. We generalize results of [3, 7] and prove them relative to ${\mathsf {N}}$ formulae, the class of forlmulae with No Nested Implications to the Left. The main application of this generalization is to characterize the provability logic of Heyting Arithmetic $\mathsf {HA}$ and prove its decidability [16].

We present a new uniform method for studying modal companions of superintuitionistic rule systems and related notions, based on the machinery of stable canonical rules. Using this method, we obtain alternative proofs of the Blok–Esakia theorem and of the Dummett–Lemmon conjecture for rule systems. Since stable canonical rules may be developed for any rule system admitting filtration, our method generalizes smoothly to richer signatures. Using essentially the same argument, we obtain a proof of an analogue of the Blok–Esakia theorem for bi-superintuitionistic and tense rule systems, and of the Kuznetsov–Muravitsky isomorphism between rule systems extending the modal intuitionistic logic $\mathtt {KM}$ and modal rule systems extending the provability logic $\mathtt {GL}$. In addition, our proof of the Dummett–Lemmon conjecture also generalizes to the bi-superintuitionistic and tense cases.

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].

Motivated by ideas from the model theory of metric structures, we introduce a metric set theory, $\mathsf {MSE}$, which takes bounded quantification as primitive and consists of a natural metric extensionality axiom (the distance between two sets is the Hausdorff distance between their extensions) and an approximate, non-deterministic form of full comprehension (for any real-valued formula $\varphi (x,y)$, tuple of parameters a, and $r < s$, there is a set containing the class and contained in the class $\{x:\varphi (x,a) < s\}$). We show that $\mathsf {MSE}$ is sufficient to develop classical mathematics after the addition of an appropriate axiom of infinity. We then construct canonical representatives of well-order types and prove that ultrametric models of $\mathsf {MSE}$ always contain externally ill-founded ordinals, conjecturing that this is true of all models. To establish several independence results and, in particular, consistency, we construct a variety of models, including pseudo-finite models and models containing arbitrarily large standard ordinals. Finally, we discuss how to formalize $\mathsf {MSE}$ in either continuous logic or Łukasiewicz logic.

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.

Let $\Omega $ be a complex lattice which does not have complex multiplication and $\wp =\wp _\Omega $ the Weierstrass $\wp $-function associated with it. Let $D\subseteq \mathbb {C}$ be a disc and $I\subseteq \mathbb {R}$ be a bounded closed interval such that $I\cap \Omega =\varnothing $. Let $f:D\rightarrow \mathbb {C}$ be a function definable in $(\overline {\mathbb {R}},\wp |_I)$. We show that if f is holomorphic on D then f is definable in $\overline {\mathbb {R}}$. The proof of this result is an adaptation of the proof of Bianconi for the $\mathbb {R}_{\exp }$ case. We also give a characterization of lattices with complex multiplication in terms of definability and a nondefinability result for the modular j-function using similar methods.

Let T be the theory of dense cyclically ordered sets with at least two elements. We determine the classifying space of $\mathsf {Mod}(T)$ to be homotopically equivalent to $\mathbb {CP}^\infty $. In particular, $\pi _2(\lvert \mathsf {Mod}(T)\rvert )=\mathbb {Z}$, which answers a question in our previous work. The computation is based on Connes’ cycle category $\Lambda $.

Cummings, Foreman, and Magidor proved that Jensen’s square principle is non-compact at $\aleph _\omega $, meaning that it is consistent that $\square _{\aleph _n}$ holds for all $n<\omega $ while $\square _{\aleph _\omega }$ fails. We investigate the natural question of whether this phenomenon generalizes to singulars of uncountable cofinality. Surprisingly, we show that under some mild ${{\mathsf {PCF}}}$-theoretic hypotheses, the weak square principle $\square _\kappa ^*$ is in fact compact at singulars of uncountable cofinality.

Consider a variant of the usual story about the iterative conception of sets. As usual, at every stage, you find all the (bland) sets of objects which you found earlier. But you also find the result of tapping any earlier-found object with any magic wand (from a given stock of magic wands).

By varying the number and behaviour of the wands, we can flesh out this idea in many different ways. This paper's main Theorem is that any loosely constructive way of fleshing out this idea is synonymous with a ZF-like theory.

This Theorem has rich applications; it realizes John Conway's (1976) Mathematicians' Liberation Movement; and it connects with a lovely idea due to Alonzo Church (1974).

In this paper, we study the employment of $\Sigma _1$-sentences with certificates, i.e., $\Sigma _1$-sentences where a number of principles is added to ensure that the witness is sufficiently number-like. We develop certificates in some detail and illustrate their use by reproving some classical results and proving some new ones. An example of such a classical result is Vaught’s theorem of the strong effective inseparability of $\mathsf {R}_0$.

We also develop the new idea of a theory being $\mathsf {R}_{0\mathsf {p}}$-sourced. Using this notion, we can transfer a number of salient results from $\mathsf {R}_0$ to a variety of other theories.

The endomorphism monoid of a model-theoretic structure carries two interesting topologies: on the one hand, the topology of pointwise convergence induced externally by the action of the endomorphisms on the domain via evaluation; on the other hand, the Zariski topology induced within the monoid by (non-)solutions to equations. For all concrete endomorphism monoids of $\omega $-categorical structures on which the Zariski topology has been analysed thus far, the two topologies were shown to coincide, in turn yielding that the pointwise topology is the coarsest Hausdorff semigroup topology on those endomorphism monoids.

We establish two systematic reasons for the two topologies to agree, formulated in terms of the model-complete core of the structure. Further, we give an example of an $\omega $-categorical structure on whose endomorphism monoid the topology of pointwise convergence and the Zariski topology differ, answering a question of Elliott, Jonušas, Mitchell, Péresse, and Pinsker.

In the study of the arithmetic degrees the $\omega \text {-REA}$ sets play a role analogous to the role the r.e. degrees play in the study of the Turing degrees. However, much less is known about the arithmetic degrees and the role of the $\omega \text {-REA}$ sets in that structure than about the Turing degrees. Indeed, even basic questions such as the existence of an $\omega \text {-REA}$ set of minimal arithmetic degree are open. This paper makes progress on this question by demonstrating that some promising approaches inspired by the analogy with the r.e. sets fail to show that no $\omega \text {-REA}$ set is arithmetically minimal. Finally, it constructs a $\prod ^0_{2}$ singleton of minimal arithmetic degree. Not only is this a result of considerable interest in its own right, constructions of $\prod ^0_{2}$ singletons often pave the way for constructions of $\omega \text {-REA}$ sets with similar properties. Along the way, a number of interesting results relating arithmetic reducibility and rates of growth are established.

In this paper, we show that $\mathsf {ZFC}+\mathsf {WA}_{n+1}$ implies the consistency of $\mathsf {ZFC}+\mathsf {WA}_n$ for $n\ge 0$. We also prove that $\mathsf {ZFC}+\mathsf {WA}_n$ is finitely axiomatizable, and $\mathsf {ZFC}+\mathsf {WA}$ is not finitely axiomatizable unless it is inconsistent.

In the first part of this paper, we consider several natural axioms in urelement set theory, including the Collection Principle, the Reflection Principle, the Dependent Choice scheme and its generalizations, as well as other axioms specifically concerning urelements. We prove that these axioms form a hierarchy over $\text {ZFCU}_{\text {R}}$ (ZFC with urelements formulated with Replacement) in terms of direct implication. The second part of the paper studies forcing over countable transitive models of $\text {ZFU}_{\text {R}}$. We propose a new definition of ${\mathbb P}$-names to address an issue with the existing approach. We then prove the fundamental theorem of forcing with urelements regarding axiom preservation. Moreover, we show that forcing can destroy and recover certain axioms within the previously established hierarchy. Finally, we demonstrate how ground model definability may fail when the ground model contains a proper class of urelements.

For many years, there have been conducting research (e.g., by Bergelson, Furstenberg, Kojman, Kubiś, Shelah, Szeptycki, Weiss) into sequentially compact spaces that are, in a sense, topological counterparts of some combinatorial theorems, for instance, Ramsey’s theorem for coloring graphs, Hindman’s finite sums theorem, and van der Waerden’s arithmetical progressions theorem. These spaces are defined with the aid of different kinds of convergences: IP-convergence, R-convergence, and ordinary convergence.

The first aim of this paper is to present a unified approach to these various types of convergences and spaces. Then, using this unified approach, we prove some general theorems about existence of the considered spaces and show that all results obtained so far in this subject can be derived from our theorems.

The second aim of this paper is to obtain new results about the specific types of these spaces. For instance, we construct a Hausdorff Hindman space that is not an $\mathcal {I}_{1/n}$-space and a Hausdorff differentially compact space that is not Hindman. Moreover, we compare Ramsey spaces with other types of spaces. For instance, we construct a Ramsey space that is not Hindman and a Hindman space that is not Ramsey.

The last aim of this paper is to provide a characterization that shows when there exists a space of one considered type that is not of the other kind. This characterization is expressed in purely combinatorial manner with the aid of the so-called Katětov order that has been extensively examined for many years so far.

This paper may interest the general audience of mathematicians as the results we obtain are on the intersection of topology, combinatorics, set theory, and number theory.

For any $n<\omega $ we construct an infinite $(n+1)$-generated Heyting algebra whose n-generated subalgebras are of cardinality $\leq m_n$ for some positive integer $m_n$. From this we conclude that for every $n<\omega $ there exists a variety of Heyting algebras which contains an infinite $(n+1)$-generated algebra, but which contains only finite n-generated algebras. For the case $n=2$ this provides a negative answer to a question posed by G. Bezhanishvili and R. Grigolia in [4].