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.

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.

We define a natural notion of standard translation for the formulas of conditional logic which is analogous to the standard translation of modal formulas into the first-order logic. We briefly show that this translation works (modulo a lightweight first-order encoding of the conditional models) for the minimal classical conditional logic $\mathsf {CK}$ introduced by Brian Chellas in [3]; however, the main result of the article is that a classically equivalent reformulation of these notions (i.e., of standard translation plus theory of conditional models) also faithfully embeds the basic Nelsonian conditional logic $\mathsf {N4CK}$, introduced in [11] into $\mathsf {QN4}$, the paraconsistent variant of Nelson’s first-order logic of strong negation. Thus $\mathsf {N4CK}$ is the logic induced by the Nelsonian reading of the classical Chellas semantics of conditionals and can, therefore, be considered a faithful analogue of $\mathsf {CK}$ on the non-classical basis provided by the propositional fragment of $\mathsf {QN4}$. Moreover, the methods used to prove our main result can be easily adapted to the case of modal logic, which makes it possible to improve an older result [10, Proposition 7] by S. Odintsov and H. Wansing about the standard translation embedding of the Nelsonian modal logic $\mathsf {FSK}^d$ into $\mathsf {QN4}$.

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 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.

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