Abstract We investigate the relationship between various versions of Ramsey’s theorem and bounding schemes in a model N $\mathcal {N}$ script upper N of a fragment of arithmetic F $F$ upper F . Our primary objective is to recast and extend the seminal results of Hirst [15] (see Theorem 1) through the modern lens of Weihrauch reducibility. By extracting explicit Weihrauch reductions from classical proofs, we expose the uniform computational content underlying these combinatorial principles, yielding a deeper and more refined understanding of their reverse mathematical strength. Our results, informally stated in our terminology and established inside N $\mathcal {N}$ script upper N , are as follows: The following are equivalent: B Σ 2 0 $\mathrm {B}\Sigma ^0_2$ normal upper B normal upper Sigma 2 Superscript 0 , the statement that the finite union of finite c.e. sets is finite, and the Infinite Pigeonhole Principle (see Theorem 3). We also discuss the Weihrauch relations between these logically equivalent principles (see Section 4). The Infinite Pigeonhole Principle is Weihrauch reducible to RT 2 2 $\mathrm {RT}^2_2$ upper R upper T 2 squared (see Theorem 4). There is also another principle logically equivalent to B Σ 2 0 $\mathrm {B}\Sigma ^0_2$ normal upper B normal upper Sigma 2 Superscript 0 which is Weihrauch reducible to SRT 2 2 $\mathrm {SRT}^2_2$ upper S upper R upper T 2 squared (see Theorem 5). We show that there is a principle which is equivalent to B Σ 3 0 $\mathrm {B}\Sigma ^0_3$ normal upper B normal upper Sigma 3 Superscript 0 (see Theorem 6) and Weihrauch reducible to SRT < ∞ 2 $\mathrm {SRT}^2_{<\infty }$ upper S upper R upper T Subscript less than infinity Superscript 2 (Theorem 7). We discuss some equivalences with B Σ n 0 $\mathrm {B}\Sigma ^0_{n}$ normal upper B normal upper Sigma Subscript n Superscript 0 (see Section 6.1) and end with a problem Weihrauch reducible to RT 2 n + 1 $\mathrm {RT}^{n+1}_{2}$ upper R upper T 2 Superscript n plus 1 (Section 6.2). Since we work within the model N $\mathcal {N}$ script upper N , many standard definitions must be adjusted. Due to the expository nature of this article, these definitions are introduced throughout the text as needed. Reading the article from start to finish will provide a better understanding of the ideas involved than focusing only on individual theorems.
In this article, we deal with the classification complexity of continuous (Devaney) chaotic systems in dimensions $0,1,$ and $\infty $ using the framework of invariant descriptive set theory. We identify the complexity in dimensions $0$ and $\infty $ , while in dimension $1$ we get some partial results.More precisely, we prove the topological conjugacy relation of invertible chaotic systems on the Hilbert cube (resp. on all compact metric spaces) has the same complexity as (i.e., is Borel bireducible with) the universal orbit relation induced by a Polish group. As a consequence, this answers a recent question asked by L. Ding. We also prove that the topological conjugacy relation of invertible chaotic systems on the Cantor space has the same complexity as the universal relation induced by the group $S_\infty $ . This answers a recent question by M. Foreman. Some non-trivial bounds on the classification complexity of chaotic systems on the interval and on the circle are also obtained. Namely, the lower bound is the Vitali equivalence relation, and the upper bound is the equality of countable sets of reals. This especially implies that the relation is Borel. However, the exact complexity remains unknown.
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 by the first author in a prior paper. This leads to new developments also in independence logic and probabilistic independence logic.
It is a well-known empirical phenomenon that natural axiomatic theories are pre-well-ordered by consistency strength. Without a precise mathematical definition of"natural,"it is unclear how to study this phenomenon mathematically. We will discuss the significance of this problem and survey some strategies that have recently been developed for addressing it. These strategies emphasize the role of reflection principles and ordinal analysis and draw on analogies with research in recursion theory. We will conclude with a discussion of open problems and directions for future research.
The Bristol model is an inner model of $L[c]$, where $c$ is a Cohen real, which is not constructible from a set. The idea was developed in 2011 in a workshop taking place in Bristol, but was only written in detail by the author in [8]. This paper is a guide for those who want to get a broader view of the construction. We try to provide more intuition that might serve as a jumping board for those interested in this construction and in odd models of $\mathsf{ZF}$. We also correct a few minor issues in the original paper, as well as prove new results. For example, that the Boolean Prime Ideal theorem fails in the Bristol model, as some sets cannot be linearly ordered, and that the ground model is always definable in its Bristol extensions. In addition to this we include a discussion on Kinna--Wagner Principles, which we think may play an important role in understanding the generic multiverse in $\mathsf{ZF}$.
The development of the forcing method has shown that several key questions regarding infinite sets cannot be settled under ZFC alone. The most widely supported view is that this undecidability simply reflects the limitations of ZFC in addressing all mathematical problems. This perspective has motivated an extensive search for new axioms—the so-called large cardinal axioms —which, when added to ZFC, yield a deeper and more robust understanding of the set-theoretic universe. Along this line, the dissertation is divided into three thematic blocks, each of them framed within these extensions of ZFC: • Very large cardinals at the threshold of Kunen inconsistency, with a focus on elementarity and cardinal correctness (Chapter 2). • Generalized Descriptive Set Theory at singular strong limit cardinals of countable cofinality, with a focus on two regularity properties (Chapter 3). • Covering lemmas and Woodin’s HOD Dichotomy through the lens of Shelah’s pcf theory (Chapter 4). Specifically, Chapter 2 establishes an inconsistency result using tools from singular cardinal combinatorics and Shelah’s pcf theory, proving the nonexistence of cardinal preserving elementary embeddings into V and establishing thereby a limitation in the hierarchy of large cardinal axioms. The proof is based on the notion of good scales and its connection with Jónsson cardinals. Chapter 3 proves a consistency result obtained via a Prikry-type forcing construction, providing a singular-cardinal analogue of Solovay’s theorem. This thematic block is inspired by Woodin’s Axiom $I_0$ which provides the appropriate axiomatic framework to develop Generalized Descriptive Set Theory in generalized Baire/Cantor spaces at singular cardinals. Concretely, we construct a model of ZFC where $\kappa $ is a strong limit singular cardinal with countable cofinality, and every subset of ${}^\omega \kappa $ in $L(V_{\kappa +1})$ has both the $\kappa $ -Perfect Set Property and the $\vec {\mathcal {U}}$ -Baire Property. Chapter 4 further explores the study of “covering lemmas” and Woodin’s HOD Dichotomy, employing both the perspective and the tools of pcf theory. Specifically, the connection between the cover property and a new pcf-theoretic concept—called the scale property —is analyzed. This analysis builds a bridge between combinatorial principles in pcf theory and the structural behavior of HOD. Abstract prepared by Sebastiano Thei. E-mail : thei91.seba@gmail.com . URL : https://air.uniud.it/handle/11390/1316344?mode=full .
This article examines the results of [6] in a purely tree-theoretic context. We show how this rephrased Chase-Freitag Lemma is sufficient to prove the Sauer-Shelah and Bhaskar Lemmas. We also use this approach to expand these two lemmas to the context of arbitrary finite label sets.
This article offers a philosophical overview and investigation of the problem of incompleteness in set theory and what this entails for the ensuing debates about proposed extensions of $ZFC$ . The incompleteness of $ZFC$ is well-known and leaves us with a rich array of competing extensions. What should we make of disagreements between them? We start by considering second-order logic and its categoricity theorems and how they might be used to compare different set theories. We then aim to use interpretability as a way of understanding that some of these debates are insubstantial. This culminates in some discussion of the relationship between interpretability and the generic multiverse. The second half of the article then takes up a more modest goal: we search for common ground and settle for partial agreement between set theories in much the same way that physicists are often content with empirical agreement. We then aim to describe a natural bound on the amount of agreement that we can expect to obtain between reasonable extensions of $ZFC$ .
We carry out a logical analysis of a convergence proof for greedy approximation schemes in uniformly smooth Banach spaces. Though the proof is by contradiction, we are able to extract computable rates of convergence that depend on the corresponding modulus of uniform smoothness for the space. While our quantitative results represent a first proof-theoretic study of greedy approximation schemes, we use this case study more generally as an opportunity to make explicit some of the high-level proof-theoretic reasoning that enables us to transform a nonconstructive convergence proof to one where computable convergence rates are apparent, representing the proof using a series of formal derivations that are designed to capture core mathematical reasoning, as opposed to low-level proof-theoretic bureaucracy. In this way we exemplify an approach to representing the process of program extraction that might, in particular, inform efforts to formalise proof mining in proof assistants.
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 $ . Theorem C. There is a model of $\mathsf {ZF}+\mathsf {DC}$ with a partition of $\mathbb {R}^3$ in unit circles but without a wellordering on the reals. 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
