Every discrete definable subset of a closed asymptotic couple with ordered scalar field ${\boldsymbol {k}}$ is shown to be contained in a finite-dimensional ${\boldsymbol {k}}$-linear subspace of that couple. It follows that the differential-valued field $\mathbb {T}$ of transseries induces more structure on its value group than what is definable in its asymptotic couple equipped with its scalar multiplication by real numbers, where this asymptotic couple is construed as a two-sorted structure with $\mathbb {R}$ as the underlying set for the second sort.

We give a new approach to the failure of the Canonical Base Property (CBP) in the so far only known counterexample, produced by Hrushovski, Palacín and Pillay. For this purpose, we will give an alternative presentation of the counterexample as an additive cover of an algebraically closed field. We isolate two fundamental weakenings of the CBP, which already appeared in work of Chatzidakis and Moosa-Pillay and show that they do not hold in the counterexample. In order to do so, a study of imaginaries in additive covers is developed. As a by-product of the presentation, we observe that a pure binding-group-theoretic account of the CBP is unlikely.

A low linear order with no computable copy constructed by C. Jockusch and R. Soare has Hausdorff rank equal to $2$. In this regard, the question arises, how simple can be a low linear order with no computable copy from the point of view of the linear order type? The main result of this work is an example of a low strong $\eta $-representation with no computable copy that is the simplest possible example.

Roelcke non-precompactness, simplicity, and non-amenability of the automorphism group of the Fraïssé limit of finite Heyting algebras are proved among others.

We introduce and prove the consistency of a new set theoretic axiom we call the Invariant Ideal Axiom. The axiom enables us to provide (consistently) a full topological classification of countable sequential groups, as well as fully characterize the behavior of their finite products.

We also construct examples that demonstrate the optimality of the conditions in IIA and list a number of open questions.

The current paper studies the formal properties of the Global Reflection Principle, to wit the assertion “All theorems of $\mathrm {Th}$ are true,” where $\mathrm {Th}$ is a theory in the language of arithmetic and the truth predicate satisfies the usual Tarskian inductive conditions for formulae in the language of arithmetic. We fix the gap in Kotlarski’s proof from [15], showing that the Global Reflection Principle for Peano Arithmetic is provable in the theory of compositional truth with bounded induction only ( $\mathrm {CT}_0$ ). Furthermore, we extend the above result showing that $\Sigma _1$ -uniform reflection over a theory of uniform Tarski biconditionals ( $\mathrm {UTB}^-$ ) is provable in $\mathrm {CT}_0$ , thus answering the question of Beklemishev and Pakhomov [2]. Finally, we introduce the notion of a prolongable satisfaction class and use it to study the structure of models of $\mathrm {CT}_0$ . In particular, we provide a new model-theoretical characterization of theories of finite iterations of uniform reflection and present a new proof characterizing the arithmetical consequences of $\mathrm {CT}_0$ .

We exhibit a bridge between the theory of cellular categories, used in algebraic topology and homological algebra, and the model-theoretic notion of stable independence. Roughly speaking, we show that the combinatorial cellular categories (those where, in a precise sense, the cellular morphisms are generated by a set) are exactly those that give rise to stable independence notions. We give two applications: on the one hand, we show that the abstract elementary classes of roots of Ext studied by Baldwin–Eklof–Trlifaj are stable and tame. On the other hand, we give a simpler proof (in a special case) that combinatorial categories are closed under 2-limits, a theorem of Makkai and Rosický.

We present a framework for tame geometry on Henselian valued fields, which we call Hensel minimality. In the spirit of o-minimality, which is key to real geometry and several diophantine applications, we develop geometric results and applications for Hensel minimal structures that were previously known only under stronger, less axiomatic assumptions. We show the existence of t-stratifications in Hensel minimal structures and Taylor approximation results that are key to non-Archimedean versions of Pila–Wilkie point counting, Yomdin’s parameterization results and motivic integration. In this first paper, we work in equi-characteristic zero; in the sequel paper, we develop the mixed characteristic case and a diophantine application.

Given a Henselian valuation, we study its definability (with and without parameters) by examining conditions on the value group. We show that any Henselian valuation whose value group is not closed in its divisible hull is definable in the language of rings, using one parameter. Thereby we strengthen known definability results. Moreover, we show that in this case, one parameter is optimal in the sense that one cannot obtain definability without parameters. To this end, we present a construction method for a t-Henselian non-Henselian ordered field elementarily equivalent to a Henselian field with a specified value group.

We introduce several highness notions on degrees related to the problem of computing isomorphisms between structures, provided that isomorphisms exist. We consider variants along axes of uniformity, inclusion of negative information, and several other problems related to computing isomorphisms. These other problems include Scott analysis (in the form of back-and-forth relations), jump hierarchies, and computing descending sequences in linear orders.

The complete characterisation of order types of non-standard models of Peano arithmetic and its extensions is a famous open problem. In this paper, we consider subtheories of Peano arithmetic (both with and without induction), in particular, theories formulated in proper fragments of the full language of arithmetic. We study the order types of their non-standard models and separate all considered theories via their possible order types. We compare the theories with and without induction and observe that the theories without induction tend to have an algebraic character that allows model constructions by closing a model under the relevant algebraic operations.

Let R be a discrete valuation domain with field of fractions Q and maximal ideal generated by $\pi $ . Let $\Lambda $ be an R-order such that $Q\Lambda $ is a separable Q-algebra. Maranda showed that there exists $k\in \mathbb {N}$ such that for all $\Lambda $ -lattices L and M, if $L/L\pi ^k\simeq M/M\pi ^k$ , then $L\simeq M$ . Moreover, if R is complete and L is an indecomposable $\Lambda $ -lattice, then $L/L\pi ^k$ is also indecomposable. We extend Maranda’s theorem to the class of R-reduced R-torsion-free pure-injective $\Lambda $ -modules.

As an application of this extension, we show that if $\Lambda $ is an order over a Dedekind domain R with field of fractions Q such that $Q\Lambda $ is separable, then the lattice of open subsets of the R-torsion-free part of the right Ziegler spectrum of $\Lambda $ is isomorphic to the lattice of open subsets of the R-torsion-free part of the left Ziegler spectrum of $\Lambda $ .

Furthermore, with k as in Maranda’s theorem, we show that if M is R-torsion-free and $H(M)$ is the pure-injective hull of M, then $H(M)/H(M)\pi ^k$ is the pure-injective hull of $M/M\pi ^k$ . We use this result to give a characterization of R-torsion-free pure-injective $\Lambda $ -modules and describe the pure-injective hulls of certain R-torsion-free $\Lambda $ -modules.

We study model theory of fields with actions of a fixed finite group scheme. We prove the existence and simplicity of a model companion of the theory of such actions, which generalizes our previous results about truncated iterative Hasse–Schmidt derivations [13] and about Galois actions [14]. As an application of our methods, we obtain a new model complete theory of actions of a finite group on fields of finite imperfection degree.

Let${\mathbb M}$ be an affine variety equipped with a foliation, both defined over a number field ${\mathbb K}$. For an algebraic $V\subset {\mathbb M}$ over ${\mathbb K}$, write $\delta _{V}$ for the maximum of the degree and log-height of V. Write $\Sigma _{V}$ for the points where the leaves intersect V improperly. Fix a compact subset ${\mathcal B}$ of a leaf ${\mathcal L}$. We prove effective bounds on the geometry of the intersection ${\mathcal B}\cap V$. In particular, when $\operatorname {codim} V=\dim {\mathcal L}$ we prove that $\#({\mathcal B}\cap V)$ is bounded by a polynomial in $\delta _{V}$ and $\log \operatorname {dist}^{-1}({\mathcal B},\Sigma _{V})$. Using these bounds we prove a result on the interpolation of algebraic points in images of ${\mathcal B}\cap V$ by an algebraic map $\Phi $. For instance, under suitable conditions we show that $\Phi ({\mathcal B}\cap V)$ contains at most $\operatorname {poly}(g,h)$ algebraic points of log-height h and degree g.

We deduce several results in Diophantine geometry. Following Masser and Zannier, we prove that given a pair of sections $P,Q$ of a nonisotrivial family of squares of elliptic curves that do not satisfy a constant relation, whenever $P,Q$ are simultaneously torsion their order of torsion is bounded effectively by a polynomial in $\delta _{P},\delta _{Q}$; in particular, the set of such simultaneous torsion points is effectively computable in polynomial time. Following Pila, we prove that given $V\subset {\mathbb C}^{n}$, there is an (ineffective) upper bound, polynomial in $\delta _{V}$, for the degrees and discriminants of maximal special subvarieties; in particular, it follows that the André–Oort conjecture for powers of the modular curve is decidable in polynomial time (by an algorithm depending on a universal, ineffective Siegel constant). Following Schmidt, we show that our counting result implies a Galois-orbit lower bound for torsion points on elliptic curves of the type previously obtained using transcendence methods by David.

Several different versions of the theory of numerosities have been introduced in the literature. Here, we unify these approaches in a consistent frame through the notion of set of labels, relating numerosities with the Kiesler field of Euclidean numbers. This approach allows us to easily introduce, by means of numerosities, ordinals and their natural operations, as well as the Lebesgue measure as a counting measure on the reals.

We initiate an investigation of structures on the set of real numbers having the property that path components of definable sets are definable. All o-minimal structures on $(\mathbb {R},<)$ have the property, as do all expansions of $(\mathbb {R},+,\cdot ,\mathbb {N})$ . Our main analytic-geometric result is that any such expansion of $(\mathbb {R},<,+)$ by Boolean combinations of open sets (of any arities) either is o-minimal or defines an isomorph of $(\mathbb N,+,\cdot )$ . We also show that any given expansion of $(\mathbb {R}, <, +,\mathbb {N})$ by subsets of $\mathbb {N}^n$ (n allowed to vary) has the property if and only if it defines all arithmetic sets. Variations arise by considering connected components or quasicomponents instead of path components.

We provide a model theoretical and tree property-like characterization of $\lambda $ - $\Pi ^1_1$ -subcompactness and supercompactness. We explore the behavior of these combinatorial principles at accessible cardinals.

We introduce and study model-theoretic connected components of rings as an analogue of model-theoretic connected components of definable groups. We develop their basic theory and use them to describe both the definable and classical Bohr compactifications of rings. We then use model-theoretic connected components to explicitly calculate Bohr compactifications of some classical matrix groups, such as the discrete Heisenberg group ${\mathrm {UT}}_3({\mathbb {Z}})$, the continuous Heisenberg group ${\mathrm {UT}}_3({\mathbb {R}})$, and, more generally, groups of upper unitriangular and invertible upper triangular matrices over unital rings.

In this paper, using a propositional modal language extended with the window modality, we capture the first-order properties of various mereological theories. In this setting, $\Box \varphi $ reads all the parts (of the current object) are $\varphi $, interpreted on the models with a whole-part binary relation under various constraints. We show that all the usual mereological theories can be captured by modal formulas in our language via frame correspondence. We also correct a mistake in the existing completeness proof for a basic system of mereology by providing a new construction of the canonical model.

The notion of a complete type can be generalized in a natural manner to allow assigning a value in an arbitrary Boolean algebra $\mathcal {B}$ to each formula. We show some basic results regarding the effect of the properties of $\mathcal {B}$ on the behavior of such types, and show they are particularity well behaved in the case of NIP theories. In particular, we generalize the third author’s result about counting types, as well as the notion of a smooth type and extending a type to a smooth one. We then show that Keisler measures are tied to certain Boolean types and show that some of the results can thus be transferred to measures—in particular, giving an alternative proof of the fact that every measure in a dependent theory can be extended to a smooth one. We also study the stable case. We consider this paper as an invitation for more research into the topic of Boolean types.