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 extend the theoretical framework of proof mining by establishing general logical metatheorems that allow for the extraction of the computational content of theorems with prima facie “noncomputational” proofs from probability theory, thereby unlocking a major branch of mathematics as a new area of application for these methods. Concretely, we first devise proof-theoretically tame logical systems that allow for the formalization of proofs involving algebras of sets together with probability contents, that is probability measures which are only assumed to be finitely additive. Based on these systems, we provide extensions for the tame treatment of Lebesgue integrals on probability contents as well as $\sigma $-algebras and associated probability measures, all via intensional approaches. All these systems are then shown to be amenable to proof-theoretic metatheorems in the style of proof mining which guarantee the extractability of effective and tame bounds from large classes of ineffective existence proofs in probability theory. Moreover, these extractable bounds are guaranteed to be highly uniform in the sense that they will be independent of all parameters relating to the underlying probability space, particularly regarding events or measures of them. As such, these results in particular provide the first logical explanation for the success and the observed uniformities of the previous ad hoc case studies of proof mining in these areas and further illustrate their extent. Lastly, we establish a general proof-theoretic transfer principle that allows for the lift of quantitative information on a relationship between different modes of convergence for sequences of real numbers to sequences of random variables.

Normal modal logics extending the logic $\mathsf {K4.3}$ of linear transitive frames are known to lack the Craig interpolation property (CIP), except some logics of bounded depth such as $\mathsf {S5}$. We turn this ‘negative’ fact into a research question and pursue a non-uniform approach to Craig interpolation by investigating the following interpolant existence problem: decide whether there exists a Craig interpolant between two given formulas in any fixed logic above $\mathsf {K4.3}$. Using a bisimulation-based characterisation of interpolant existence for descriptive frames, we show that this problem is decidable and coNP-complete for all finitely axiomatisable normal modal logics containing $\mathsf {K4.3}$. It is thus not harder than entailment in these logics, which is in sharp contrast to other recent non-uniform interpolation results. We also extend our approach to Priorean temporal logics (with both past and future modalities) over the standard time flows—the integers, rationals, reals, and finite strict linear orders—none of which is blessed with the CIP.

We introduce the notion of a weak A2 space (or wA2-space), which generalises spaces satisfying Todorčević’s axioms A1–A4 and countable vector spaces. We show that in any Polish weak A2 space, analytic sets are Kastanas Ramsey, and discuss the relationship between Kastanas Ramsey sets and sets in the projective hierarchy. We also show that in all spaces satisfying A1–A4, every subset of $\mathcal {R}$ is Kastanas Ramsey iff Ramsey, generalising the recent result by [2]. Finally, we show that in the setting of Gowers wA2-spaces, Kastanas Ramsey sets and strategically Ramsey sets coincide, providing a connection between the recent studies on topological Ramsey spaces and countable vector spaces.

In this note, we prove that Kim-dividing over models is always witnessed by a coheir Morley sequence, whenever the theory is NATP.

Following the strategy of Chernikov and Kaplan [8], we obtain some corollaries which hold in NATP theories. Namely, (i) if a formula Kim-forks over a model, then it quasi-divides over the same model and (ii) for any tuple of parameters b and a model M, there exists a global coheir p containing $\text {tp}(b/M)$ such that for all $b'\models p|_{MB}$.

We also show that for coheirs in NATP theories, condition (ii) above is a necessary condition for being a witness of Kim-dividing, assuming that a witness of Kim-dividing exists (see Definition 4.1 in this note). That is, if we assume that a witness of Kim-dividing always exists over any given model, then a coheir $p\supseteq \text {tp}(a/M)$ must satisfy (ii) whenever it is a witness of Kim-dividing of a over a model M. We also give a sufficient condition for the existence of a witness of Kim-dividing in terms of pre-independence relations.

At the end of the article, we leave a short remark on Mutchnik’s recent work [17]. We point out that the class of N-$\omega $-DCTP$_2$ theories, a subclass of the class of NATP theories, contains all NTP$_2$ theories and NSOP$_1$ theories. We also note that Kim-forking and Kim-dividing are equivalent over models in N-$\omega $-NDCTP$_2$ theories, where Kim-dividing is defined with respect to invariant Morley sequences, instead of coheir Morley sequences as in [17].

This study focuses on certain combinations of rules or conditions involving a would-be ‘provability’ or ‘truth’ predicate that would render a system of arithmetic containing them either straightforwardly inconsistent (if those predicates were assumed to be definable) or logico-semantically paradoxical (if those predicates were taken as primitive and governed by the rules in question). These two negative properties are not to be conflated; we conjecture, however, that they are complementary. Logico-semantic paradoxicality, we contend, admits of proof-theoretic analysis: the ‘disproofs’ involved do not reveal straightforward inconsistency. This is because, unlike the disproofs involved in establishing straightforward inconsistencies, these paradox-revealing ‘disproofs’ cannot be brought into normal form.

The border between metamathematical proofs of certain (constructive) impossibility results and the non-normalizable (and always constructive) disproofs engendered by semantic paradoxicality is not fully understood. The respective strategies of reasoning on each side—genuine proofs of inconsistency versus whatever kind of ‘disproof’ uncovers semantic paradoxicality—seem somehow similar. They seem to involve the same ‘lines of reasoning’. But there is an important and principled difference between them.

This difference will be emphasized throughout our discussion of certain arithmetical impossibility results, and closely related semantic paradoxes. The proof-theoretic criterion for paradoxicality is that in the case of paradoxes (as opposed to genuine inconsistencies) the apparent ‘disproofs’ that use the rules stipulated for the primitive predicates in question cannot be brought into normal form. In proof-theoretic terminology: their reduction sequences do not terminate. This means that cut fails for languages generating paradox. But cut holds for the language of arithmetic. It follows that the paradox-generating primitive predicates of a semantically closed language cannot be defined in arithmetical terms. For, if they could be, then they could be replaced by their definitions within the paradoxical disproofs, and the resulting disproofs would be normalizable.

This article is concerned with finite rank stability theory, and more precisely two classical ways to decompose a type using minimal types. The first is its domination equivalence to a Morley product of minimal types, and the second is its semi-minimal analysis, both of which are useful in applications. Our main interest is to explore how these two decompositions are connected. We prove that neither determine the other in general, and give more precise connections using various notions from the model theory literature such as uniform internality, proper fibrations, and disintegratedness.

We consider the problem of predicting the next bit in an infinite binary sequence sampled from the Cantor space with an unknown computable measure. We propose a new theoretical framework to investigate the properties of good computable predictions, focusing on such predictions’ convergence rate.

Since no computable prediction can be the best, we first define a better prediction as one that dominates the other measure. We then prove that this is equivalent to the condition that the sum of the KL divergence errors of its predictions is smaller than that of the other prediction for more computable measures. We call that such a computable prediction is more general than the other.

We further show that the sum of any sufficiently general prediction errors is a finite left-c.e. Martin-Löf random real. This means the errors converge to zero more slowly than any computable function.

The paper proposes and studies new classical, type-free theories of truth and determinateness with unprecedented features. The theories are fully compositional, strongly classical (namely, their internal and external logics are both classical), and feature a defined determinateness predicate satisfying desirable and widely agreed principles. The theories capture a conception of truth and determinateness according to which the generalizing power associated with the classicality and full compositionality of truth is combined with the identification of a natural class of sentences—the determinate ones—for which clear-cut semantic rules are available. Our theories can also be seen as the classical closures of Kripke–Feferman truth: their $\omega $-models, which we precisely pin down, result from including in the extension of the truth predicate the sentences that are satisfied by a Kripkean closed-off fixed-point model. The theories compare to recent theories proposed by Fujimoto and Halbach, featuring a primitive determinateness predicate. In the paper we show that our theories entail all principles of Fujimoto and Halbach’s theories, and are proof-theoretically equivalent to Fujimoto and Halbach’s $\mathsf {CD}^{+}$. We also show establish some negative results on Fujimoto and Halbach’s theories: such results show that, unlike what happens in our theories, the primitive determinateness predicate prevents one from establishing clear and unrestricted semantic rules for the language with type-free truth.

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.

Within the determinacy setting, ${\mathscr {P}({\omega _1})}$ is regular (in the sense of cofinality) with respect to many known cardinalities and thus there is substantial evidence to support the conjecture that ${\mathscr {P}({\omega _1})}$ has globally regular cardinality. However, there is no known information about the regularity of ${\mathscr {P}(\omega _2)}$. It is not known if ${\mathscr {P}(\omega _2)}$ is even $2$-regular under any determinacy assumptions. The article will provide the following evidence that ${\mathscr {P}(\omega _2)}$ may possibly be ${\omega _1}$-regular: Assume $\mathsf {AD}^+$. If $\langle A_\alpha : \alpha < {\omega _1} \rangle $ is such that ${\mathscr {P}(\omega _2)} = \bigcup _{\alpha < {\omega _1}} A_\alpha $, then there is an $\alpha < {\omega _1}$ so that $\neg (|A_\alpha | \leq |[\omega _2]^{<\omega _2}|)$.

We build on a 1990 paper of Bukovský and Copláková-Hartová. First, we remove the hypothesis of ${\mathsf {CH}}$ from one of their minimality results. Then, using a measurable cardinal, we show that there is a $|\aleph _2^V|=\aleph _1$-minimal extension that is not a $|\aleph _3^V|=\aleph _1$-extension, answering the first of their questions.

Building on the correspondence between finitely axiomatised theories in Łukasiewicz logic and rational polyhedra, we prove that the unification type of the fragment of Łukasiewicz logic with $n\geqslant 2$ variables is nullary. This solves a problem left open by V. Marra and L. Spada [Ann. Pure Appl. Logic 164 (2013), pp. 192–210]. Furthermore, we refine the study of unification with bounds on the number of variables. Our proposal distinguishes the number m of variables allowed in the problem and the number n in the solution. We prove that the unification type of Łukasiewicz logic for all $m,n \geqslant 2$ is nullary.

This article is a contribution to the “neostability” type of result for abstract elementary classes. Under certain set theoretic assumptions, we propose a definition and a characterization of NIP in AECs. The class of AECs with NIP properly contains the class of stable AECs.1 We show that for an AEC K and $\lambda \geq LS(K)$, $K_\lambda $ is NIP if and only if there is a notion of nonforking on it which we call a w*-good frame. On the other hand, the negation of NIP leads to being able to encode subsets.

We prove that the satisfaction relation $\mathcal {N}\models \varphi [\vec a]$ of first-order logic is not absolute between models of set theory having the structure $\mathcal {N}$ and the formulas $\varphi $ all in common. Two models of set theory can have the same natural numbers, for example, and the same standard model of arithmetic $\left \langle {\mathbb N},{+},{\cdot },0,1, <\right \rangle $, yet disagree on their theories of arithmetic truth; two models of set theory can have the same natural numbers and the same arithmetic truths, yet disagree on their truths-about-truth, at any desired level of the iterated truth-predicate hierarchy; two models of set theory can have the same natural numbers and the same reals, yet disagree on projective truth; two models of set theory can have the same $\left \langle {H}_{\omega _2},{\in }\right \rangle $ or the same rank-initial segment $\left \langle {V}_\delta ,{\in }\right \rangle $, yet disagree on which assertions are true in these structures.

On the basis of these mathematical results, we argue that a philosophical commitment to the determinateness of the theory of truth for a structure cannot be seen as a consequence solely of the determinateness of the structure in which that truth resides. The determinate nature of arithmetic truth, for example, is not a consequence of the determinate nature of the arithmetic structure ${\mathbb N}=\{\,{0,1,2,\ldots }\,\}$ itself, but rather, we argue, is an additional higher-order commitment requiring its own analysis and justification.

A longstanding question is to characterize the lattice of supersets (modulo finite sets), $\mathcal {L}^*(A)$, of a low$_2$ computably enumerable (c.e.) set. The conjecture is that $\mathcal {L}^*(A)\cong {\mathcal E}^*$. In spite of claims in the literature, this longstanding question/conjecture remains open. We contribute to this problem by solving one of the main test cases. We show that if c.e. A is low$_2$ then A has an atomless hyperhypersimple superset. In fact, if A is c.e. and low$_2$, then for any $\Sigma _3$-Boolean algebra B there is some c.e. $H\supseteq A$ such that $\mathcal {L}^*(H)\cong B$.

Logical inferentialists have expected identity to be susceptible of harmonious introduction and elimination rules in natural deduction. While Read and Klev have proposed rules they argue are harmonious, Griffiths and Ahmed have criticized these rules as insufficient for harmony. These critics, moreover, suggest that no harmonious rules are forthcoming. We argue that these critics are correct: the logical inferentialist should abandon hope for harmonious rules for identity. The paper analyzes the three major uses of identity in presumed-logical languages: variable coordination, definitional substitution, and co-reference. We show that identity qua variable coordination is not logical by providing a harmonious natural-deduction system that captures this use through the quantifiers. We then argue that identity qua definitional substitution or co-reference faces a dilemma: either its rules are harmonious but they obscure its actual use in inference, or its rules are not harmonious but they make its actual use in inference plain. We conclude that the inferentialist may have harmonious rules for identity only by disrespecting its inferential use.

We show that the class of Krasner hyperfields is not elementary. To show this, we determine the rational rank of quotients of multiplicative groups in field extensions. We also discuss some related questions.

For any $2 \le n < \omega $, we introduce a forcing poset using generalized promises which adds a normal n-splitting subtree to a $(\ge \! n)$-splitting normal Aronszajn tree. Using this forcing poset, we prove several consistency results concerning finitely splitting subtrees of Aronszajn trees. For example, it is consistent that there exists an infinitely splitting Suslin tree whose topological square is not Lindelöf, which solves an open problem due to Marun. For any $2 < n < \omega $, it is consistent that every $(\ge \! n)$-splitting normal Aronszajn tree contains a normal n-splitting subtree, but there exists a normal infinitely splitting Aronszajn tree which contains no $(< \! n)$-splitting subtree. To show the latter consistency result, we prove a forcing iteration preservation theorem related to not adding new small-splitting subtrees of Aronszajn trees.