A Polish space is not always homeomorphic to a computably presented Polish space. In this article, we examine degrees of non-computability of presenting homeomorphic copies of compact Polish spaces. We show that there exists a $\mathbf {0}'$-computable low$_3$ compact Polish space which is not homeomorphic to a computable one, and that, for any natural number $n\geq 2$, there exists a Polish space $X_n$ such that exactly the high$_{n}$-degrees are required to present the homeomorphism type of $X_n$. Along the way we investigate the computable aspects of Čech homology groups. We also show that no compact Polish space has a least presentation with respect to Turing reducibility.

I continue the study of the blurry HOD hierarchy. The technically most involved result is that the theory ZFC + “$\aleph _\omega $ is a strong limit cardinal and $\aleph _{\omega +1}$ is the least leap” is equiconsistent with the theory ZFC + “there is a measurable cardinal.”

We construct divergent models of $\mathsf {AD}^+$ along with the failure of the Continuum Hypothesis ($\mathsf {CH}$) under various assumptions. Divergent models of $\mathsf {AD}^+$ play an important role in descriptive inner model theory; all known analyses of HOD in $\mathsf {AD}^+$ models (without extra iterability assumptions) are carried out in the region below the existence of divergent models of $\mathsf {AD}^+$. Our results are the first step toward resolving various open questions concerning the length of definable prewellorderings of the reals and principles implying $\neg \mathsf {CH}$, like $\mathsf {MM}$, that divergent models shed light on, see Question 5.1.

We generalise the properties $\mathsf {OP}$, $\mathsf {IP}$, k-$\mathsf {TP}$, $\mathsf {TP}_{1}$, k-$\mathsf {TP}_{2}$, $\mathsf {SOP}_{1}$, $\mathsf {SOP}_{2}$, and $\mathsf {SOP}_{3}$ to positive logic, and prove various implications and equivalences between them. We also provide a characterisation of stability in positive logic in analogy with the one in full first-order logic, both on the level of formulas and on the level of theories. For simple theories there are the classically equivalent definitions of not having $\mathsf {TP}$ and dividing having local character, which we prove to be equivalent in positive logic as well. Finally, we show that a thick theory T has $\mathsf {OP}$ iff it has $\mathsf {IP}$ or $\mathsf {SOP}_{1}$ and that T has $\mathsf {TP}$ iff it has $\mathsf {SOP}_{1}$ or $\mathsf {TP}_{2}$, analogous to the well-known results in full first-order logic where $\mathsf {SOP}_{1}$ is replaced by $\mathsf {SOP}$ in the former and by $\mathsf {TP}_{1}$ in the latter. Our proofs of these final two theorems are new and make use of Kim-independence.

A terminating sequent calculus for intuitionistic propositional logic is obtained by modifying the R$\supset $ rule of the labelled sequent calculus $\mathbf {G3I}$. This is done by adding a variant of the principle of a fortiori in the left-hand side of the premiss of the rule. In the resulting calculus, called ${\mathbf {G3I}}_{\mathbf {t}}$, derivability of any given sequent is directly decidable by root-first proof search, without any extra device such as loop-checking. In the negative case, the failed proof search gives a finite countermodel to the sequent on a reflexive, transitive, and Noetherian Kripke frame. As a byproduct, a direct proof of faithfulness of the embedding of intuitionistic logic into Grzegorcyk logic is obtained.

We study cofinal systems of finite subsets of $\omega _1$. We show that while such systems can be NIP, they cannot be defined in an NIP structure. We deduce a positive answer to a question of Chernikov and Simon from 2013: In an NIP theory, any uncountable externally definable set contains an infinite definable subset. A similar result holds for larger cardinals.

The Lovász local lemma (LLL) is a technique from combinatorics for proving existential results. There are many different versions of the LLL. One of them, the lefthanded local lemma, is particularly well suited for applications to two player games. There are also constructive and computable versions of the LLL. The chief object of this thesis is to prove an effective version of the lefthanded local lemma and to apply it to effectivise constructions of non-repetitive sequences.

The second goal of this thesis is to categorize some classes of cohesive powers. We completely describe both the isomorphism types of cohesive powers of equivalence structures and injection structures, as well as clarify the relationship between these cohesive powers and their original structures. We also describe the finite condensation of cohesive powers of computable copies of the integers as a linear order by cohesive sets whose complement are computably enumerable.

Finally, we investigate the possibility of decomposing problems in the Weihrauch degrees into a product of first order part and second order part. We give a preliminary result in this direction.

Abstract prepared by Daniel Mourad.

E-mail: daniel.mourad@uconn.edu

URL: https://daniel-mourad.scholar.uconn.edu/wp-content/uploads/sites/2530/2023/06/ThesisFinalWithRevisions-2.pdf

Computability theoretic aspects of Polish metric spaces are studied by adapting notions and methods of computable structure theory. In this dissertation, we mainly investigate index sets and classification problems for computably presentable Polish metric spaces. We find the complexity of a number of index sets, isomorphism problems, and embedding problems for computably presentable metric spaces. We also provide several computable structure theory results related to some classical Polish metric spaces such as the Urysohn space $\mathbb {U}$, the Cantor space $2^{\mathbb {N}}$, the Baire space $\mathbb {N}^{\mathbb {N}}$, and spaces of continuous functions.

Abstract prepared by Teerawat Thewmorakot.

E-mail: teerawat.thew@hotmail.com

There is a balance between the amount of (weak) indestructibility one can have and the amount of strong cardinals. It’s consistent relative to large cardinals to have lots of strong cardinals and all of their degrees of strength are weakly indestructible. But this necessitates the destructibility of the partially strong cardinals. Guaranteeing the indestructibility of the partially strong cardinals is shown to be harder. In particular, this work establishes an equiconsistency between:

1. 1. a proper class of cardinals that are strong reflecting strongs; and

2. 2. weak indestructibility for (κ+2)-strength for all cardinals κ in the presence of a proper class of strong cardinals.

1. 3. weak indestructibility for all degrees of strength for a proper class of strong cardinals.

One direction of the equiconsistency of (1) and (2) is proven using forcing and the other using core model techniques from inner model theory. Additionally, connections between weak indestructibility and the reflection properties associated with Woodin cardinals are discussed, and similar results are derived for supercompacts and supercompacts reflecting supercompacts.

Abstract prepared by James Holland.

E-mail: jch258@scarletmail.rutgers.edu

URL: https://sites.math.rutgers.edu/~jch258/assets/dis.pdf

We study ordered groups in the context of both algebra and computability. Ordered groups are groups that admit a linear order that is compatible with the group operation. We explore some properties of ordered groups and discuss some related topics. We prove results about the semidirect product in relation to orderability and computability. In particular, we give a criteria for when a semidirect product of orderable groups is orderable and for when a semidirect product is computably categorical. We also give an example of a semidirect product that has the halting set coded into its multiplication structure but it is possible to construct a computable presentation of this semidirect product.

We examine a family of orderable groups that admit exactly countably many orders and show that their space of orders has arbitrary finite Cantor–Bendixson rank. Furthermore, this family of groups is also shown to be computably categorical, which in particular will allow us to conclude that any computable presentation of the groups does not admit any noncomputable orders. Lastly, we construct an example of an orderable computable group with no computable Archimedean orders but at least one computable non-Archimedean order.

Abstract prepared by Waseet Kazmi.

E-mail: waseet.kazmi@uconn.edu

URL: http://hdl.handle.net/11134/20002:860745910

We study the complexities of isometry and isomorphism classes of separable Banach spaces in the Polish spaces of Banach spaces, recently introduced and investigated by the authors in [14]. We obtain sharp results concerning the most classical separable Banach spaces.

We prove that the infinite-dimensional separable Hilbert space is characterized as the unique separable infinite-dimensional Banach space whose isometry class is closed, and also as the unique separable infinite-dimensional Banach space whose isomorphism class is $F_\sigma $. For $p\in \left [1,2\right )\cup \left (2,\infty \right )$, we show that the isometry classes of $L_p[0,1]$ and $\ell _p$ are $G_\delta $-complete sets and $F_{\sigma \delta }$-complete sets, respectively. Then we show that the isometry class of $c_0$ is an $F_{\sigma \delta }$-complete set.

Additionally, we compute the complexities of many other natural classes of separable Banach spaces; for instance, the class of separable $\mathcal {L}_{p,\lambda +}$-spaces, for $p,\lambda \geq 1$, is shown to be a $G_\delta $-set, the class of superreflexive spaces is shown to be an $F_{\sigma \delta }$-set, and the class of spaces with local $\Pi $-basis structure is shown to be a $\boldsymbol {\Sigma }^0_6$-set. The paper is concluded with many open problems and suggestions for a future research.

Ultrafilters play a significant role in model theory to characterize logics having various compactness and interpolation properties. They also provide a general method to construct extensions of first-order logic having these properties. A main result of this paper is that every class $\Omega $ of uniform ultrafilters generates a $\Delta $-closed logic ${\mathcal {L}}_\Omega $. ${\mathcal {L}}_\Omega $ is $\omega $-relatively compact iff some $D\in \Omega $ fails to be $\omega _1$-complete iff ${\mathcal {L}}_\Omega $ does not contain the quantifier “there are uncountably many.” If $\Omega $ is a set, or if it contains a countably incomplete ultrafilter, then ${\mathcal {L}}_\Omega $ is not generated by Mostowski cardinality quantifiers. Assuming $\neg 0^\sharp $ or $\neg L^{\mu }$, if $D\in \Omega $ is a uniform ultrafilter over a regular cardinal $\nu $, then every family $\Psi $ of formulas in ${\mathcal {L}}_\Omega $ with $|\Phi |\leq \nu $ satisfies the compactness theorem. In particular, if $\Omega $ is a proper class of uniform ultrafilters over regular cardinals, ${\mathcal {L}}_\Omega $ is compact.

An infinite graph is said to be highly connected if the induced subgraph on the complement of any set of vertices of smaller size is connected. We continue the study of weaker versions of Ramsey’s theorem on uncountable cardinals asserting that if we color edges of the complete graph, we can find a large highly connected monochromatic subgraph. In particular, several questions of Bergfalk, Hrušák, and Shelah (2021, Acta Mathematica Hungarica 163, 309–322) are answered by showing that assuming the consistency of suitable large cardinals, the following are relatively consistent with ZFC:

• • $\kappa \to _{hc} (\kappa )^2_\omega $ for every regular cardinal $\kappa \geq \aleph _2$,

• • $\neg \mathsf {CH}+ \aleph _2 \to _{hc} (\aleph _1)^2_\omega $.

Building on a work of Lambie-Hanson (2023, Fundamenta Mathematicae. 260(2):181–197), we also show that

• • $\aleph _2 \to _{hc} [\aleph _2]^2_{\omega ,2}$ is consistent with $\neg \mathsf {CH}$.

To prove these results, we use the existence of ideals with strong combinatorial properties after collapsing suitable large cardinals.

I provide simplified proofs for each of the following fundamental theorems regarding selection principles:

1. (1) The Quasinormal Convergence Theorem, due to the author and Zdomskyy, asserting that a certain, important property of the space of continuous functions on a space is actually preserved by Borel images of that space.

2. (2) The Scheepers Diagram Last Theorem, due to Peng, completing all provable implications in the diagram.

3. (3) The Menger Game Theorem, due to Telgársky, determining when Bob has a winning strategy in the game version of Menger’s covering property.

4. (4) A lower bound on the additivity of Rothberger’s covering property, due to Carlson.

The simplified proofs lead to several new results.

We construct an existentially undecidable complete discretely valued field of mixed characteristic with existentially decidable residue field and decidable algebraic part, answering a question by Anscombe–Fehm in a strong way. Along the way, we construct an existentially decidable field of positive characteristic with an existentially undecidable finite extension, modifying a construction due to Kesavan Thanagopal.

Cook and Reckhow [5] pointed out that $\mathcal {N}\mathcal {P} \neq co\mathcal {N}\mathcal {P}$ iff there is no propositional proof system that admits polynomial size proofs of all tautologies. The theory of proof complexity generators aims at constructing sets of tautologies hard for strong and possibly for all proof systems. We focus on a conjecture from [16] in foundations of the theory that there is a proof complexity generator hard for all proof systems. This can be equivalently formulated (for p-time generators) without a reference to proof complexity notions as follows:

• • There exists a p-time function g stretching each input by one bit such that its range $rng(g)$ intersects all infinite $\mathcal {N}\mathcal {P}$ sets.

$\bigvee $

$\bigvee $

$\mathcal {N}\mathcal {P}$

$\mathcal {N}\mathcal {P}$

Given an uncountable cardinal $\kappa $, we consider the question of whether subsets of the power set of $\kappa $ that are usually constructed with the help of the axiom of choice are definable by $\Sigma _1$-formulas that only use the cardinal $\kappa $ and sets of hereditary cardinality less than $\kappa $ as parameters. For limits of measurable cardinals, we prove a perfect set theorem for sets definable in this way and use it to generalize two classical nondefinability results to higher cardinals. First, we show that a classical result of Mathias on the complexity of maximal almost disjoint families of sets of natural numbers can be generalized to measurable limits of measurables. Second, we prove that for a limit of countably many measurable cardinals, the existence of a simply definable well-ordering of subsets of $\kappa $ of length at least $\kappa ^+$ implies the existence of a projective well-ordering of the reals. In addition, we determine the exact consistency strength of the nonexistence of $\Sigma _1$-definitions of certain objects at singular strong limit cardinals. Finally, we show that both large cardinal assumptions and forcing axioms cause analogs of these statements to hold at the first uncountable cardinal $\omega _1$.

We introduce and study a class of betweenness algebras—Boolean algebras with binary operators, closely related to ternary frames with a betweenness relation. From various axioms for betweenness, we chose those that are most common, which makes our work applicable to a wide range of betweenness structures studied in the literature. On the algebraic side, we work with two operators of possibility and of sufficiency.

We introduce and study the model-theoretic notions of absolute connectedness and type-absolute connectedness for groups. We prove that groups of rational points of split semisimple linear groups (that is, Chevalley groups) over arbitrary infinite fields are absolutely connected and characterize connected Lie groups which are type-absolutely connected. We prove that the class of type-absolutely connected group is exactly the class of discretely topologized groups with the trivial Bohr compactification, that is, the class of minimally almost periodic groups.

Purity is known as an ideal of proof that restricts a proof to notions belonging to the ‘content’ of the theorem. In this paper, our main interest is to develop a conception of purity for formal (natural deduction) proofs. We develop two new notions of purity: one based on an ontological notion of the content of a theorem, and one based on the notions of surrogate ontological content and structural content. From there, we characterize which (classical) first-order natural deduction proofs of a mathematical theorem are pure. Formal proofs that refer to the ontological content of a theorem will be called ‘fully ontologically pure’. Formal proofs that refer to a surrogate ontological content of a theorem will be called ‘secondarily ontologically pure’, because they preserve the structural content of a theorem. We will use interpretations between theories to develop a proof-theoretic criterion that guarantees secondary ontological purity for formal proofs.