As a continuation of the work of the third author in [5], we make further observations on the features of Galois cohomology in the general model theoretic context. We make explicit the connection between forms of definable groups and first cohomology sets with coefficients in a suitable automorphism group. We then use a method of twisting cohomology (inspired by Serre’s algebraic twisting) to describe arbitrary fibres in cohomology sequences—yielding a useful “finiteness” result on cohomology sets.Applied to the special case of (...) differential fields and Kolchin’s constrained cohomology, we complete results from [3] by proving that the first constrained cohomology set of a differential algebraic group over a bounded, differentially large, field is countable. (shrink)

We show that if a countable structure M in a finite relational language is not cellular, then there is an age-preserving $N \supseteq M$ such that $2^{\aleph _0}$ many structures are bi-embeddable with N. The proof proceeds by a case division based on mutual algebraicity.

The thesis is divided into two parts. The first one focuses on generalized descriptive set theory, and the second one on combinatorics, model theory, and Ramsey theory. Generalized descriptive set theory (GDST) is a natural extension of (classical) descriptive set theory (DST) where countable is replaced by uncountable. But the framework of GDST is narrow if compared to that of DST, as so far GDST has mostly concentrated on the study of the generalized Baire space, rather than considering arbitrary “Polish-like” (...) spaces or standard $\kappa $ -Borel spaces. Also, GDST is usually developed for cardinals satisfying $\kappa ^{ \omega $. All the proposed classes give rise to the same class of spaces up to $ \kappa $ -Borel isomorphism, providing a natural setup to work with. Then, various results from classical DST about Polish, Borel, and standard Borel spaces are extended to this context. Chapter 2 extends the previous analysis to embrace singular cardinals too. In particular, it contains an in-depth study of the generalizations and characterizations of metrizability necessary in the singular case. The main result on this topic is a new metrization theorem in terms of topological games that holds for both classical metrizability and $\mathbb {G}$ -metrizability. Chapter 3 features various examples of spaces in the classes considered above, and a study of linearly ordered topological spaces (LOTS) and generalized ordered spaces (GO-spaces) in the context of GDST. The second part of the thesis deals with a recently discovered notion in combinatorics. In 2019, Solecki introduced the classes of Ramsey monoids and $\mathbb {Y}$ -controllable monoids to collect and extend different theorems in combinatorics, like Hindman’s Finite Sum Theorem, Carlson’s Theorem, Gowers’ FIN $_\kappa $ Theorem, and Furstenberg–Katznelson’s Ramsey Theorem. Then, he provided a necessary condition and some sufficient conditions for a finite monoid to be Ramsey or $\mathbb {Y}$ -controllable. Chapters 4 and 5 aim to continue the work started by Solecki on these and other related classes of monoids. We improve the necessary conditions and the sufficient conditions provided by Solecki, reaching in particular a full characterization of Ramsey monoids. This further extends results like Carlson’s Theorem and Gowers’ FIN $_\kappa $ Theorem, but it also sets a precise limit on when it is possible to obtain similar statements. We also give examples of classes of $\mathbb {Y}$ -controllable monoids that do not satisfy some of the sufficient conditions, suggesting possible strategies to improve the results we provided. Then, we show that in certain particular classes of $\mathbb {Y}$ -controllable monoids with stronger properties, the remaining sufficient conditions become necessary as well. In Chapter 5, we also study local versions of the classes of Ramsey and $\mathbb {Y}$ -controllable monoids that are better suited for infinite monoids. The thesis contains material from joint works with Luca Motto Ros, Philipp Schlicht, and Eugenio Colla. Abstract prepared by Claudio Agostini E-mail: [email protected] Current affiliation: HUN-REN ALFRÉD RÉNYI INSTITUTE OF MATHEMATICS REÁLTANODA UTCA 13-15 H-1053, BUDAPEST. (shrink)

We introduce and study the model-theoretic notions of absolute connectedness and type-absolute connectedness for groups. We prove that groups of rational points of split semisimple linear groups (that is, Chevalley groups) over arbitrary infinite fields are absolutely connected and characterize connected Lie groups which are type-absolutely connected. We prove that the class of type-absolutely connected group is exactly the class of discretely topologized groups with the trivial Bohr compactification, that is, the class of minimally almost periodic groups.

In this article, we study the commutativity between the pull-back and the push-forward functors on constructible functions in Cluckers–Loeser motivic integration.

We explore a general method based on trees of elementary submodels in order to present highly simplified proofs to numerous results in infinite combinatorics. While countable elementary submodels have been employed in such settings already, we significantly broaden this framework by developing the corresponding technique for countably closed models of size continuum. The applications range from various theorems on paradoxical decompositions of the plane, to coloring sparse set systems, results on graph chromatic number and constructions from point-set topology. Our main (...) purpose is to demonstrate the ease and wide applicability of this method in a form accessible to anyone with a basic background in set theory and logic. (shrink)

We give a unified overview of the study of the effects of additional set theoretic axioms on quotient structures. Our focus is on rigidity, measured in terms of existence (or rather non-existence) of suitably non-trivial automorphisms of the quotients in question. A textbook example for the study of this topic is the Boolean algebra $\mathcal {P}({\mathbb N})/\operatorname {\mathrm {Fin}}$, whose behavior is the template around which this survey revolves: Forcing axioms imply that all of its automorphisms are trivial, in the (...) sense that they are induced by almost permutations of ${\mathbb N}$, while under the Continuum Hypothesis this rigidity fails and $\mathcal {P}({\mathbb N})/\operatorname {\mathrm {Fin}}$ admits uncountably many non-trivial automorphisms. We consider far-reaching generalisations of this phenomenon and present a wide variety of situations where analogous patterns persist, focusing mainly (but not exclusively) on the categories of Boolean algebras, Čech–Stone remainders, and $\mathrm {C}^{*}$ -algebras. We survey the state of the art and the future prospects of this field, discussing the major open problems and outlining the main ideas of the proofs whenever possible. (shrink)

Traditionally, the role of general topology in model theory has been mainly limited to the study of compacta that arise in first-order logic. In this context, the topology tends to be so trivial that it turns into combinatorics, motivating a widespread approach that focuses on the combinatorial component while usually hiding the topological one. This popular combinatorial approach to model theory has proved to be so useful that it has become rare to see more advanced topology in model-theoretic articles. Prof. (...) Franklin D. Tall has led the re-introduction of general topology as a valuable tool to push the boundaries of model theory. Most of this thesis is directly influenced by and builds on this idea.The first part of the thesis will answer a problem of T. Gowers on the undefinability of pathological Banach spaces such as Tsirelson space. The topological content of this chapter is centred around Grothendieck spaces.In a similar spirit, the second part will show a new connection between the notion of metastability introduced by T. Tao and the topological concept of pseudocompactness. We shall make use of this connection to show a result of X. Caicedo, E. Dueñez, J. Iovino in a much simplified manner.The third part of the thesis will carry a higher set-theoretic content as we shall use forcing and descriptive set theory to show that the well-known theorem of M. Morley on the trichotomy concerning the number of models of a first-order countable theory is undecidable if one considers second-order countable theories instead.The only part that did not originate from model-theoretic questions will be the fourth one. We show that $\operatorname {ZF} + \operatorname {DC} +$ “all Turing invariant sets of reals have the perfect set property” implies that all sets of reals have the perfect set property. We also show that this result generalizes to all countable analytic equivalence relations. This result provides evidence in favour of a long-standing conjecture asking whether Turing determinacy implies the axiom of determinacy.Abstract prepared by Clovis HamelE-mail: [email protected]. (shrink)

The original Specker–Blatter theorem (1983) was formulated for classes of structures $\mathcal {C}$ of one or several binary relations definable in Monadic Second Order Logic MSOL. It states that the number of such structures on the set $[n]$ is modularly C-finite (MC-finite). In previous work we extended this to structures definable in CMSOL, MSOL extended with modular counting quantifiers. The first author also showed that the Specker–Blatter theorem does not hold for one quaternary relation (2003).If the vocabulary allows a constant (...) symbol c, there are n possible interpretations on $[n]$ for c. We say that a constant c is hard-wired if c is always interpreted by the same element $j \in [n]$. In this paper we show: (i)The Specker–Blatter theorem also holds for CMSOL when hard-wired constants are allowed. The proof method of Specker and Blatter does not work in this case.(ii)The Specker–Blatter theorem does not hold already for $\mathcal {C}$ with one ternary relation definable in First Order Logic FOL. This was left open since 1983.Using hard-wired constants allows us to show MC-finiteness of counting functions of various restricted partition functions which were not known to be MC-finite till now. Among them we have the restricted Bell numbers $B_{r,A}$, restricted Stirling numbers of the second kind $S_{r,A}$ or restricted Lah-numbers $L_{r,A}$. Here r is a non-negative integer and A is an ultimately periodic set of non-negative integers. (shrink)

We study properties of definable sets and functions in power-bounded T -convex fields, proving that the latter have the multidimensional Jacobian property and that the theory of T -convex fields is b -minimal with centers. Through these results and work of I. Halupczok we ensure that a certain kind of geometrical stratifications exist for definable objects in said fields. We then discuss a number of applications of those stratifications, including applications to Archimedean o-minimal geometry.

A sequence of graphs is FO-convergent if the probability of satisfaction of every first-order formula converges. A graph modeling is a graph, whose domain is a standard probability space, with the property that every definable set is Borel. It was known that FO-convergent sequence of graphs do not always admit a modeling limit, but it was conjectured that FO-convergent sequences of sufficiently sparse graphs have a modeling limits. Precisely, two conjectures were proposed: 1. If a FO-convergent sequence of graphs is (...) residual, that is if for every integer d the maximum relative size of a ball of radius d in the graphs of the sequence tends to zero, then the sequence has a modeling limit. 2. A monotone class of graphs ${\cal C}$ has the property that every FO-convergent sequence of graphs from ${\cal C}$ has a modeling limit if and only if ${\cal C}$ is nowhere dense, that is if and only if for each integer p there is $N\left( p \right)$ such that no graph in ${\cal C}$ contains the pth subdivision of a complete graph on $N\left( p \right)$ vertices as a subgraph. In this article we prove both conjectures. This solves some of the main problems in the area and among others provides an analytic characterization of the nowhere dense–somewhere dense dichotomy. (shrink)

We use the model completeness and axiomatisation of the theory of closed ordered differential fields to give a differential version of Dubois, Krivine and Risler's nullstellensatz and Stengle's positivstellensatz for ordered fields.

In a recent article, Freitag, Moosa, and the author showed that in differentially closed fields of characteristic zero, if two types are nonorthogonal, then their n+3 and m+3 Morley powers are not weakly orthogonal, where n and m are their respective Lascar ranks. In this short note, we prove that the bound is tight: There are such types with weakly orthogonal n+2 and m+2 Morley powers. The types in question were constructed by Freitag and Moosa as examples of types with (...) degree of nonminimality 2. As interesting as our result are our methods: We rely mostly on Galois theory and some descent argument for types, combined with the failure of the inverse Galois problem over constant parameters. (shrink)

In this note we prove and disprove some chain conditions in type definable and definable groups in dependent, strongly dependent and strongly2 dependent theories.

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. (shrink)

The isomorphism invariance criterion of logical nature has much to commend it. It can be philosophically motivated by the thought that logic is distinctively general or topic neutral. It is capable of precise set-theoretic formulation. And it delivers an extension of ‘logical constant’ which respects the intuitively clear cases. Despite its attractions, the criterion has recently come under attack. Critics such as Feferman, MacFarlane and Bonnay argue that the criterion overgenerates by incorrectly judging mathematical notions as logical. We consider five (...) possible precisifications of the overgeneration argument and find them all unconvincing. (shrink)

Given a rough definably amenable rough approximate subgroup A of a group in some first-order structure there is a type-definable subgroup N normalized by A and contained in A4 of bounded index in the subgroup ⟨A⟩ generated by A.

Inquisitive modal logic, InqML, in its epistemic incarnation, extends standard epistemic logic to capture not just the information that agents have, but also the questions that they are interested in. We use the natural notion of bisimulation equivalence in the setting of InqML, as introduced in [7], to characterise the expressiveness of InqML as the bisimulation invariant fragment of first-order logic over natural classes of two-sorted first-order structures that arise as relational encodings of inquisitive epistemic (S5-like) models. The non-elementary nature (...) of these classes crucially requires non-classical model-theoretic methods for the analysis of first-order expressiveness, irrespective of whether we aim for characterisations in the sense of classical or of finite model theory. (shrink)

Inspired by Adler’s idea on VC minimal theories [1], we introduce VC-minimal complexity. We show that for any $N\in \mathbb {N}^{>0}$, there is $k_N>0$ such that for any finite bipartite graph $(X,Y;E)$ with VC-minimal complexity $< N$, there exist $X'\subseteq X$, $Y'\subseteq Y$ with $|X'|\geq k_N |X|$, $|Y'|\geq k_N |Y|$ such that $X'\times Y' \subseteq E$ or $X'\times Y'\cap E=\emptyset $.

We investigate the notion of a semi-retraction between two first-order structures (in typically different signatures) that was introduced by the second author as a link between the Ramsey property and generalized indiscernible sequences. We look at semi-retractions through a new lens establishing transfers of the Ramsey property and finite Ramsey degrees under quite general conditions that are optimal as demonstrated by counterexamples. Finally, we compare semi-retractions to the category theoretic notion of a pre-adjunction.

We start by showing how to approximate unitary and bounded self-adjoint operators by operators in finite dimensional spaces. Using ultraproducts we give a precise meaning for the approximation. In this process we see how the spectral measure is obtained as an ultralimit of counting measures that arise naturally from the finite dimensional approximations. Then we see how generalized distributions can be interpreted in the ultraproduct. Finally we study how one can calculate kernels of operators K by calculating them in the (...) finite dimensional approximations and how one needs to interpret Dirac deltas in the ultraproduct in order to get the kernels as propagators $\langle x_{1}|K|x_{0}\rangle $. (shrink)

The languages of finitary and infinitary logic over the alphabet of bounded lattices have proven to be of considerable use in the study of compacta. Significant among the sentences of these languages are the ones that are base free, those whose truth is unchanged when we move among the lattice bases of a compactum. In this paper we define syntactically the expansive sentences, and show each of them to be base free. We also show that many well-known properties of compacta (...) may be expressed using expansive sentences; and that any property so expressible is closed under inverse limits and co-existential images. As a byproduct, we conclude that co-existential images of pseudo-arcs are pseudo-arcs. This is of interest because the corresponding statement for confluent maps is still open, and co-existential maps are often—but not always—confluent. (shrink)

We examine topological dynamical systems on the Cantor set from the point of view of the continuous model theory of commutative C*-algebras. After some general remarks, we focus our attention on the generic homeomorphism of the Cantor set, as constructed by Akin, Glasner, and Weiss. We show that this homeomorphism is the prime model of its theory. We also show that the notion of “generic” used by Akin, Glasner, and Weiss is distinct from the notion of “generic” encountered in Fraïssé (...) theory. (shrink)

We describe an Aldous–Hoover-type characterization of random relational structures that are exchangeable relative to a fixed structure which may have various equivalence relations. Our main theorem gives the common generalization of the results on relative exchangeability due to Ackerman \)-invariant measures: part I, 2015. arXiv:1509.06170) and Crane and Towsner and hierarchical exchangeability results due to Austin and Panchenko :809–823, 2014).

Arrow’s axiomatic foundation of social choice theory can be understood as an application of Tarski’s methodology of the deductive sciences—which is closely related to the latter’s foundational contribution to model theory. In this note we show in a model-theoretic framework how Arrow’s use of von Neumann and Morgenstern’s concept of winning coalitions allows to exploit the algebraic structures involved in preference aggregation; this approach entails an alternative indirect ultrafilter proof for Arrow’s dictatorship result. This link also connects Arrow’s seminal result (...) to key developments and concepts in the history of model theory, notably ultraproducts and preservation results. (shrink)

Erdős proved that for every infinite \ there is \ with \, such that all pairs of points from Y have distinct distances, and he gave partial results for general a-ary volume. In this paper, we search for the strongest possible canonization results for a-ary volume, making use of general model-theoretic machinery. The main difficulty is for singular cardinals; to handle this case we prove the following. Suppose T is a stable theory, \ is a finite set of formulas of (...) T, \, and X is an infinite subset of M. Then there is \ with \ and an equivalence relation E on Y with infinitely many classes, each class infinite, such that Y is \\)-indiscernible. We also consider the definable version of these problems, for example we assume \ is perfect and we find some perfect \ with all distances distinct. Finally we show that Erdős’s theorem requires some use of the axiom of choice. (shrink)

Cerreia-Vioglio et al. :341–375, 2011) have proposed a very general axiomatisation of preferences in the presence of ambiguity, viz. Monotonic Bernoullian Archimedean preference orderings. This paper investigates the problem of Arrovian aggregation of such preferences—and proves dictatorial impossibility results for both finite and infinite populations. Applications for the special case of aggregating expected-utility preferences are given. A novel proof methodology for special aggregation problems, based on model theory, is employed.

A relational structure \ is called reversible iff each bijective homomorphism from \ onto \ is an isomorphism, and linear orders are prototypical examples of such structures. One way to detect new reversible structures of a given relational language L is to notice that the maximal or minimal elements of isomorphism-invariant sets of interpretations of the language L on a fixed domain X determine reversible structures. We isolate certain syntactical conditions providing that a satisfiable \-theory defines a class of interpretations (...) having extreme elements on a fixed domain and detect several classes of reversible structures. For some of these classes, we characterize the corresponding reversible extreme interpretations. In particular, we characterize the reversible countable ultrahomogeneous graphs. (shrink)