We introduce the notion of echeloned spaces – an order-theoretic abstraction of metric spaces. The first step is to characterize metrizable echeloned spaces. It turns out that morphisms between metrizable echeloned spaces are uniformly continuous or have a uniformly discrete image. In particular, every automorphism of a metrizable echeloned space is uniformly continuous, and for every metric space with midpoints, the automorphisms of the induced echeloned space are precisely the dilations.

Next, we focus on finite echeloned spaces. They form a Fraïssé class, and we describe its Fraïssé-limit both as the echeloned space induced by a certain homogeneous metric space and as the result of a random construction. Building on this, we show that the class of finite ordered echeloned spaces is Ramsey. The proof of this result combines a combinatorial argument by Nešetřil and Hubička with a topological-dynamical point of view due to Kechris, Pestov and Todorčević. Finally, using the method of Katětov functors due to Kubiś and Mašulović, we prove that the full symmetric group on a countable set topologically embeds into the automorphism group of the countable universal homogeneous echeloned space.

We prove that the class of separably algebraically closed valued fields equipped with a distinguished Frobenius endomorphism $x \mapsto x^q$ is decidable, uniformly in q. The result is a simultaneous generalization of the work of Chatzidakis and Hrushovski (in the case of the trivial valuation) and the work of the first author and Hrushovski (in the case where the fields are algebraically closed).

The logical setting for the proof is a model completeness result for valued fields equipped with an endomorphism $\sigma $ which is locally infinitely contracting and fails to be onto. Namely, we prove the existence of a model complete theory $\widetilde {\mathrm {VFE}}$ amalgamating the theories $\mathrm {SCFE}$ and $\widetilde {\mathrm {VFA}}$ introduced in [5] and [11], respectively. In characteristic zero, we also prove that $\widetilde {\mathrm {VFE}}$ is NTP$_2$ and classify the stationary types: they are precisely those orthogonal to the fixed field and the value group.

Using motivic integration theory and the notion of riso-triviality, we introduce two new objects in the framework of definable nonarchimedean geometry: a convenient partial preorder $\preccurlyeq$ on the set of constructible motivic functions, and an invariant $V_0$, nonarchimedean substitute for the number of connected components. We then give several applications based on $\preccurlyeq$ and $V_0$: we obtain the existence of nonarchimedean substitutes of real measure geometric invariants $V_i$, called the Vitushkin variations, and we establish the nonarchimedean counterpart of a real inequality involving $\preccurlyeq$, the metric entropy and our invariants $V_i$. We also prove the nonarchimedean Cauchy–Crofton formula for definable sets of dimension $d$, relating $V_0$ (and $V_d$) and the motivic measure in dimension $d$.

A first-order expansion of $(\mathbb {R},+,<)$ is dp-minimal if and only if it is o-minimal. We prove analogous results for algebraic closures of finite fields, p-adic fields, ordered abelian groups with only finitely many convex subgroups (in particular archimedean ordered abelian groups), and abelian groups equipped with archimedean cyclic group orders. The latter allows us to describe unary definable sets in dp-minimal expansions of $(\mathbb {Z},+,S)$, where S is a cyclic group order. Along the way we describe unary definable sets in dp-minimal expansions of ordered abelian groups. In the last section we give a canonical correspondence between dp-minimal expansions of $(\mathbb {Q},+,<)$ and o-minimal expansions ${\mathscr R}$ of $(\mathbb {R},+,<)$ such that $({\mathscr R},\mathbb {Q})$ is a “dense pair.”.

Pre-H-fields are ordered valued differential fields satisfying some basic axioms coming from transseries and Hardy fields. We study pre-H-fields that are differential-Hensel–Liouville closed, that is, differential-henselian, real closed, and closed under exponential integration, establishing an Ax–Kochen/Ershov theorem for such structures: the theory of a differential-Hensel–Liouville closed pre-H-field is determined by the theory of its ordered differential residue field; this result fails if the assumption of closure under exponential integration is dropped. In a two-sorted setting with one sort for a differential-Hensel–Liouville closed pre-H-field and one sort for its ordered differential residue field, we eliminate quantifiers from the pre-H-field sort, from which we deduce that the ordered differential residue field is purely stably embedded and if it has NIP, then so does the two-sorted structure. Similarly, the one-sorted theory of differential-Hensel–Liouville closed pre-H-fields with closed ordered differential residue field has quantifier elimination, is the model completion of the theory of pre-H-fields with gap $0$, and is complete, distal, and locally o-minimal.

Building on our previous work on enriched universal algebra, we define a notion of enriched language consisting of function and relation symbols whose arities are objects of the base of enrichment $\mathcal {V}$. In this context, we construct atomic formulas and define the regular fragment of our enriched logic by taking conjunctions and existential quantification of those. We then characterize $\mathcal {V}$-categories of models of regular theories as enriched injectivity classes in the $\mathcal {V}$-category of structures. These notions rely on the choice of an orthogonal factorization system $(\mathcal {E},\mathcal {M})$ on $\mathcal {V}$ which will be used, in particular, to interpret relation symbols and existential quantification.

We prove that any compact R-analytic group is linear when R is a pro-p domain of characteristic zero.

The concept of stability has proved very useful in the field of Banach space geometry. In this note, we introduce and study a corresponding concept in the setting of Banach algebras, which we call multiplicative stability. As we shall prove, various interesting examples of Banach algebras are multiplicatively unstable, and hence unstable in the model-theoretic sense. The examples include Fourier algebras over noncompact amenable groups, $C^*$-algebras and the measure algebra of an infinite compact group.

The purpose of this paper is to investigate forcing as a tool to construct universal models. In particular, we look at theories of initial segments of the universe and show that any model of a sufficiently rich fragment of those theories can be embedded into a model constructed by forcing. Our results rely on the model-theoretic properties of good ultrafilters, for which we provide a new existence proof on non-necessarily complete Boolean algebras.

We investigate the end extendibility of models of arithmetic with restricted elementarity. By utilizing the restricted ultrapower construction in the second-order context, for each $n\in \mathbb {N}$ and any countable model of $\mathrm {B}\Sigma _{n+2}$, we construct a proper $\Sigma _{n+2}$-elementary end extension satisfying $\mathrm {B}\Sigma _{n+1}$, which answers a question by Clote positively. We also give a characterization of the countable models of $\mathrm {I}\Sigma _{n+2}$ in terms of their end extendibility, similar to the case of $\mathrm {B}\Sigma _{n+2}$. Along the proof, we introduce a new type of regularity principle in arithmetic called the weak regularity principle, which serves as a bridge between the model’s end extendibility and the amount of induction or collection it satisfies.

There are known characterisations of several fragments of hybrid logic by means of invariance under bisimulations of some kind. The fragments include $\{\mathord {\downarrow }, \mathord {@}\}$ with or without nominals (Areces, Blackburn, Marx), $\mathord {@}$ with or without nominals (ten Cate), and $\mathord {\downarrow }$ without nominals (Hodkinson, Tahiri). Some pairs of these characterisations, however, are incompatible with one another. For other fragments of hybrid logic no such characterisations were known so far. We prove a generic bisimulation characterisation theorem for all standard fragments of hybrid logic, in particular for the case with $\mathord {\downarrow }$ and nominals, left open by Hodkinson and Tahiri. Our characterisation is built on a common base and for each feature extension adds a specific condition, so it is modular in an engineering sense.

We show that if a field A is not pseudo-finite, then there is no prime model of the theory of pseudo-finite fields over A. Assuming GCH, we extend this result to $\kappa $-prime models, for $\kappa $ an uncountable cardinal or $\aleph _\varepsilon $.

We study H-structures associated with $SU$-rank 1 measurable structures. We prove that the $SU$-rank of the expansion is continuous and that it is uniformly definable in terms of the parameters of the formulas. We also introduce notions of dimension and measure for definable sets in the expansion and prove they are uniformly definable in terms of the parameters of the formulas.

Let $\operatorname {TFAb}_r$ be the class of torsion-free abelian groups of rank r, and let $\operatorname {FD}_r$ be the class of fields of characteristic $0$ and transcendence degree r. We compare these classes using various notions. Considering the Scott complexity of the structures in the classes and the complexity of the isomorphism relations on the classes, the classes seem very similar. Hjorth and Thomas showed that the $\operatorname {TFAb}_r$ are strictly increasing under Borel reducibility. This is not so for the classes $\operatorname {FD}_r$. Thomas and Velickovic showed that for sufficiently large r, the classes $\operatorname {FD}_r$ are equivalent under Borel reducibility. We try to compare the groups with the fields, using Borel reducibility, and also using some effective variants. We give functorial Turing computable embeddings of $\operatorname {TFAb}_r$ in $\operatorname {FD}_r$, and of $\operatorname {FD}_r$ in $\operatorname {FD}_{r+1}$. We show that under computable countable reducibility, $\operatorname {TFAb}_1$ lies on top among the classes we are considering. In fact, under computable countable reducibility, isomorphism on $\operatorname {TFAb}_1$ lies on top among equivalence relations that are effective $\Sigma _3$.

This paper studies which truth-values are most likely to be taken on finite models by arbitrary sentences of a many-valued predicate logic. The classical zero-one law (independently proved by Fagin and Glebskiĭ et al.) states that every sentence in a purely relational language is almost surely false or almost surely true, meaning that the probability that the formula is true in a randomly chosen finite structures of cardinal n is asymptotically $0$ or $1$ as n grows to infinity. We obtain generalizations of this result for any logic with values in a finite lattice-ordered algebra, and for some infinitely valued logics, including Łukasiewicz logic. The finitely valued case is reduced to the classical one through a uniform translation and Oberschelp’s generalization of the zero-one law. Moreover, it is shown that the complexity of determining the almost sure value of a given sentence is PSPACE-complete (generalizing Grandjean’s result for the classical case), and for some logics we describe completely the set of truth-values that can be taken by sentences almost surely.

Lyubetsky and Kanovei showed in [8] that there is a second-order arithmetic model of $\mathrm {Z}_2^{-p}$, (comprehension for all second-order formulas without parameters), in which $\Sigma ^1_2$-$\mathrm {CA}$ (comprehension for all $\Sigma ^1_2$-formulas with parameters) holds, but $\Sigma ^1_4$-$\mathrm {CA}$ fails. They asked whether there is a model of $\mathrm {Z}_2^{-p}+\Sigma ^1_2$-$\mathrm {CA}$ with the optimal failure of $\Sigma ^1_3$-$\mathrm {CA}$. We answer the question positively by constructing such a model in a forcing extension by a tree iteration of Jensen’s forcing. Let $\mathrm {Coll}^{-p}$ be the parameter-free collection scheme for second-order formulas and let $\mathrm {AC}^{-p}$ be the parameter-free choice scheme. We show that there is a model of $\mathrm {Z}_2^{-p}+\mathrm { AC}^{-p}+\Sigma ^1_2$-$\mathrm {CA}$ with a failure of $\Sigma ^1_4$-$\mathrm {CA}$. We also show that there is a model of $\mathrm {Z}_2^{-p}+\mathrm {Coll}^{-p}+\Sigma ^1_2$-$\mathrm {CA}$ with a failure of $\Sigma ^1_4$-$\mathrm {CA}$ and a failure of $\mathrm {AC}^{-p}$, so that, in particular, the schemes $\mathrm {Coll}^{-p}$ and $\mathrm {AC}^{-p}$ are not equivalent over $\mathrm {Z}_2^{-p}$.

We investigate and compare applications of the Zilber–Pink conjecture and dynamical methods to rigidity problems for arithmetic real and complex hyperbolic lattices. Along the way, we obtain new general results about reconstructing a variation of Hodge structure from its typical Hodge locus that may be of independent interest. Applications to Siu’s immersion problem are also discussed, the most general of which only requires the hypothesis that infinitely many closed geodesics map to proper totally geodesic subvarieties under the immersion.

We explore the interplay between $\omega $-categoricity and pseudofiniteness for groups, and we conjecture that $\omega $-categorical pseudofinite groups are finite-by-abelian-by-finite. We show that the conjecture reduces to nilpotent p-groups of class 2, and give a proof that several of the known examples of $\omega $-categorical p-groups satisfy the conjecture. In particular, we show by a direct counting argument that for any odd prime p the ($\omega $-categorical) model companion of the theory of nilpotent class 2 exponent p groups, constructed by Saracino and Wood, is not pseudofinite, and that an $\omega $-categorical group constructed by Baudisch with supersimple rank 1 theory is not pseudofinite. We also survey some scattered literature on $\omega $-categorical groups over 50 years.

The lattice problem for models of Peano Arithmetic ($\mathsf {PA}$) is to determine which lattices can be represented as lattices of elementary submodels of a model of $\mathsf {PA}$, or, in greater generality, for a given model $\mathcal {M}$, which lattices can be represented as interstructure lattices of elementary submodels $\mathcal {K}$ of an elementary extension $\mathcal {N}$ such that $\mathcal {M}\preccurlyeq \mathcal {K}\preccurlyeq \mathcal {N}$. The problem has been studied for the last 60 years and the results and their proofs show an interesting interplay between the model theory of PA, Ramsey style combinatorics, lattice representation theory, and elementary number theory. We present a survey of the most important results together with a detailed analysis of some special cases to explain and motivate a technique developed by James Schmerl for constructing elementary extensions with prescribed interstructure lattices. The last section is devoted to a discussion of lesser-known results about lattices of elementary submodels of countable recursively saturated models of PA.

Feferman proved in 1962 [6] that any arithmetical theorem is a consequence of a suitable transfinite iteration of full uniform reflection of $\mathsf {PA}$. This result is commonly known as Feferman’s completeness theorem. The purpose of this paper is twofold. On the one hand this is an expository paper, giving two new proofs of Feferman’s completeness theorem that, we hope, shed light on this mysterious and often overlooked result. On the other hand, we combine one of our proofs with results from computable structure theory due to Ash and Knight to give sharp bounds on the order types of well-orders necessary to attain the completeness for levels of the arithmetical hierarchy.