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 ^{<\kappa }=\kappa $, which implies that $\kappa $ must be regular.

The goal of the first part of the thesis is to fill these gaps, studying classes of spaces that could take the role of Polish spaces in the generalized context under the weak assumption $2^{<\kappa }=\kappa $, which allows one to include singular cardinals and can consistently hold at every cardinal (e.g., in models of $ \mathsf {ZFC+GCH} $).

In Chapter 1, we begin by considering the case when $\kappa $ is regular. We consider several candidates for “Polish-like” spaces that have been proposed in the literature (e.g., $\mathbb {G}$-Polish spaces and $\mathrm {SC}_\kappa $-spaces), and introduce a new one ($f\mathrm {SC}_\kappa $-spaces). We show that all these classes are nicely organized in four groups, with two clear dividing lines between them: $ \kappa $-additivity, which can be interpreted as a strong analog of zero-dimensionality, and the degree of completeness (Figure 1).Figure 1

Relationships among Polish-like classes of regular Hausdorff spaces of weight $\leq \kappa $, for a totally ordered Abelian group $\mathbb {G}$ of degree $\deg (\mathbb {G})=\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: agostini.claudio@renyi.hu

Current affiliation: HUN-REN ALFRÉD RÉNYI INSTITUTE OF MATHEMATICS REÁLTANODA UTCA 13-15 H-1053, BUDAPEST

The thesis is thematically divided into two parts: Algebraic closures of certain subfields of the reals (Part I) and Paradoxical sets of reals (Part II). Part I investigates a folklore result about the forcing extension by one Cohen real: the transcendence degree of the reals over the set of reals in the ground model is of cardinality $\mathfrak {c}$ (in the extension). We extend this to the case in which more Cohen reals are added, obtaining the following result:Theorem A.

Let X be a finite set of mutually generic Cohen reals over V. In $V[X]$, consider the minimum field $F\subseteq \mathbb {R}$ such that $F\supseteq \bigcup _{Y\subsetneq X} \mathbb {R}^{V[Y]}$. Then, in $V[X]$ the transcendence degree of $\mathbb {R}$ over F is continuum.

In Part II, we consider some paradoxical sets of reals and study their interaction with the Axiom of Choice. Informally, paradoxical sets are subsets of $\mathbb {R}^n$ that can be constructed using the Axiom of Choice. In this work we focus on the following examples of paradoxical sets: Hamel bases of $\mathbb {R}$ as a $\mathbb {Q}$-vector space, two-point sets or Mazurkiewicz sets, and partitions of $\mathbb {R}^3$ into unit circles (PUC). The known proofs of existence of these objects rely on a transfinite induction on a well-order of the reals.

The main question considered through this work is the following: Can we recover some weakening of the Axiom of Choice from the existence of a particular paradoxical set? This thesis gives negative answers for different versions of this question, changing the particular paradoxical set, and the weakening of the Axiom of Choice considered. Furthermore, the main contribution of this thesis is the development of a framework that produces some of these answers and recovers other known results of similar form. In particular we obtain the following applications:Theorem B.

There is a model of $\mathsf {ZF}+\mathsf {DC}$ with a Hamel basis and no free ultrafilter on $\omega $.

Abstract prepared by Azul Fatalini

E-mail: A.L.Fatalini@leeds.ac.uk

URL: https://nbn-resolving.org/urn:nbn:de:hbz:6-53948487632

Current affiliation: UNIVERSITY OF LEEDS

We introduce finite support iterations of symmetric systems, and use them to provide a strongly modernized proof of David Pincus’ classical result that the axiom of dependent choice is independent over $\operatorname {\mathrm {ZF}}$ with the ordering principle together with a failure of the axiom of choice.

A recursive set of formulas of first-order logic with finitely many predicate letters, including “=”, has a model over the integers in which the predicates are Boolean combinations of recursively enumerable sets, if it has an infinite model at all. The proof corrects a fallacious argument published by Hensel and Putnam in 1969.

We study dependence and independence concepts found in quantum physics, especially those related to hidden variables and non-locality, through the lens of team semantics and probabilistic team semantics, adapting a relational framework introduced in [1]. This also leads to new developments in independence logic and probabilistic independence logic.

Carnap’s (Categoricity) Problem concerns the relationship between (rules of) inference and model-theoretic values. In particular, it asks whether proof-theoretic constraints are ‘strong enough’ to uniquely determine intended semantic values. Carnap [20] demonstrated that already in the classical bivalent setting this is not the case for the majority of the usual logical constants. To remedy this underdetermination of ‘semantics by syntax’ a variety of solution strategies has been explored in the literature. This article is a philosophical-logical survey of these attempts, comparing them with respect to scope, motivation, and success. Besides the mathematical interest held by Carnap’s Problem, the underdetermination it uncovers has significant consequences for a variety of philosophical projects and positions, warranting a systematic study of attempts at resolving it.

An étale structure over a topological space X is a continuous family of structures (in some first-order language) indexed over X. We give an exposition of this fundamental concept from sheaf theory and its relevance to countable model theory and invariant descriptive set theory. We show that many classical aspects of spaces of countable models can be naturally framed and generalized in the context of étale structures, including the Lopez-Escobar theorem on invariant Borel sets, an omitting types theorem, and various characterizations of Scott rank. We also present and prove the countable version of the Joyal–Tierney representation theorem, which states that the isomorphism groupoid of an étale structure determines its theory up to bi-interpretability; and we explain how special cases of this theorem recover several recent results in the literature on groupoids of models and functors between them.

In this paper, a proof-theoretic perspective on counterfactual inference is proposed. On this perspective, proof-theoretic structure is fundamental. We start from a certain primacy of inferential practice and structural proof theory. Models are required neither for the explanation of the meaning of counterfactuals, nor for that of counterfactual inference. Taking a proof-theoretic perspective and an intuitionistic stance on meaning (cf. BHK), we define modal intuitionistic natural deduction systems for drawing conclusions from counterfactual assumptions. These proof systems are modal insofar as derivations in them make use of assumption modes which are sensitive to the factuality status (e.g., factual, counterfactual) of the formula that is to be assumed. This status is determined by a reference proof system on top of which a modal proof system is defined. The rules of a modal system draw on this status.

The main results obtained are preservation, normalization, subexpression (incl. subformula) property, and internal completeness. The systems are applied to the analysis of reasoning with natural language constructions such as ‘If A were the case, B would [might] be the case’, ‘Since A is the case, B is [might be] the case’. A proof-theoretic semantics is provided for them.

A Constructive Logic of Evidence and Truth ($\mathsf {LET_C}$) is introduced. This logic is both paraconsistent and paracomplete, providing connectives for consistency and determinedness that enable the independent recovery of explosiveness and the law of excluded middle for specific propositions. Dual connectives for inconsistency and undeterminedness are also defined in $\mathsf {LET_C}$. Evidence is explicitly formalised by integrating lambda calculus terms into $\mathsf {LET_C}$, resulting in the type system $\mathsf {LET_C^{\lambda }}$. In this system, lambda calculus terms represent procedures for constructing evidence for compound formulas based on the evidence of their constituent parts. A realisability interpretation is provided for $\mathsf {LET_C}$, establishing a strong connection between deductions in this system and recursive functions.

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.

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.