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 article 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, the Boolean Prime Ideal theorem fails in the Bristol model, as some sets cannot be linearly ordered, and 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 $\mathsf {ZF}$.

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.

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.

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.

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.

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 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.

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

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 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.

The original Mitchell forcing was discovered by William Mitchell in 1974 and used by him to prove that it is consistent that there are no $\aleph _2$-Aronszajn trees, i.e., trees of height $\aleph _2$ and width $\aleph _1$ without cofinal branches. In recent years, the technique underlying Mitchell forcing has been generalized by John Krueger and used to distinguish the variants of internal approachability which were introduced by Matthew Foreman and Stevo Todorcevic—given an elementary substructure $N\prec H(\Theta )$ of size $\mu ^+$, we say that N is internally unbounded, internally stationary, or internally club depending on the largeness of $[N]^{<\mu }\cap N$ inside of $[N]^{<\mu }$ and that N is internally approachable if it can be written as the union of an increasing sequence which has all of its initial segments lying in N.

In this thesis, we generalize Mitchell’s forcing in a different way and construct variants of it to answer open questions due to Krueger himself and other authors.

In the first part of the thesis, we construct several useful tools: In Chapter 3, we introduce the concept of strong ${<\mu }$-distributivity (asserting that every $\mu $-sequence of open dense sets has a descending thread) and prove both a strengthening of Easton’s lemma concerning the interaction between the chain condition and distributivity as well as a variant of Foreman’s theorem concerning a connection between the completeness game and distributivity. In Chapter 4, we cast Mitchell forcing and its variants in an abstract way as partial orders on products of sets satisfying specific regularity properties and name a new sufficient condition for Hamkins’ approximation property. This is applied in Chapter 6, where we construct a general variant of Mitchell forcing and prove it to have various properties. In the intermediate Chapter 5, we prove the consistency of the existences of Laver guessing functions for various large cardinals.

In the second part of the thesis, we apply all of these tools to answer various open questions from the literature: In Chapter 7, we focus on the variants of internal approachability. Answering questions of Krueger, we prove that, consistently, the notions of internal stationarity and clubness as well as internal clubness and approachability can be distinct for structures of size $\aleph _n$ for all $n\geq 1$ simultaneously and, moreover, this distinction can be obtained from merely a countable sequence of Mahlo cardinals. Answering questions of Foreman and Todorcevic, we prove that it is consistent to have structures N such that, for some $\nu $, $N\cap H(\nu )$ is internally approachable but N itself is not even internally club. Lastly, we answer a question of Levine by showing that the distinction between internal stationarity and internal approachability for $\mu ^+$-sized elementary substructures of $H(\mu ^+)$ is not unconditionally equivalent to the existence of a disjoint stationary sequence on $\mu ^+$—this statement, introduced by Krueger, asserts the existence of a sequence $(\mathcal {S}_{\alpha })_{\alpha \in S}$ where $S\subseteq \mu ^+\cap \operatorname {cof}(\mu )$ and every $\mathcal {S}_{\alpha }\subseteq [\alpha ]^{<\mu }$ is stationary and is equivalent to having a distinction between internal stationarity and internal approachability for $\mu $-sized substructures of $H(\mu ^+)$ assuming $2^{\mu }=\mu ^+$. In Chapter 8, we answer open questions due to Mohammadpour regarding the ineffable slender property, a strengthening of the tree property due to Weiß. For example, we prove that the ineffable slender property can consistently be indestructible under directed-closed forcing (similar to the Laver preparation for supercompact cardinals).

Abstract prepared by Hannes Jakob.

E-mail: hannes.jakob@unt.edu.

URL: https://freidok.uni-freiburg.de/data/262351.

As suggested by the title, this thesis consists of two parts: profinite groups, and models of Presburger arithmetic. Profinite groups are covered in chapters 2–4, and chapters 5–7 cover Presburger arithmetic. Chapter 1 contains a brief introduction to computability and computable structure theory.

In Chapter 2, we give an overview of profinite groups and of tree presentations. We describe how to represent size continuum profinite groups using countable tree presentations and discuss past results about the computability of profinite groups from the literature. Chapter 3 examines the possible complexities of first-order theories of profinite groups. In particular, we find upper bounds for both the existential and positive first-order theories. We distinguish the class of profinite groups that are isomorphic to a direct product of finite groups, for which we get tighter upper bounds on the complexity of existential theories as well as upper bounds for the complexity of entire first-order theories. Additionally, we demonstrate that these bounds are sharp. In Chapter 4, we examine certain countable subgroups of profinite groups. Given a Turing ideal I and a profinite group G, we define $G_I$ to be the subgroup of all elements in G whose Turing degree lies in I. We examine under what conditions and to what extent $G_I$ will be an elementary subgroup of G. In particular, we show that there exists a profinite group whose subgroup of computable elements is not elementary even for existential formulas.

Chapter 5 provides an overview of Presburger arithmetic as well as some historical notes. We discuss divisible ordered abelian groups and their connection to Presburger groups. Additionally, we examine Presburger groups that are generated by a single nontrivial generator and show that almost all of these groups are rigid. In Chapter 6, we look at the possible degree spectra of Presburger groups. We show how given a countable linear order $\mathcal {L}$, one can construct a countable Presburger group $P_{\mathcal {L}}$ such that the degree spectrum of $P_{\mathcal {L}}$ is the set of Turing degrees whose jump is in the degree spectrum of $\mathcal {L}$. This yields many interesting degree spectra for Presburger groups. Chapter 7 deals with the possible Scott ranks of Presburger groups. We show that the Scott spectrum of Presburger arithmetic is complete. That is, given any countable ordinal $\alpha>1$, there exists a Presburger group whose Scott rank is $\alpha $. This further distinguishes Presburger arithmetic from Peano arithmetic, as no model of Peano arithmetic can have a finite Scott rank other than $1$.

Abstract prepared by Jason Block.

E-mail: jeblock@wm.edu.

URL: https://sites.google.com/view/jasonblockmath/.