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Java Darleen Villano. Computable Categoricity, and Topology in Reverse Mathematics. University of Connecticut. 2025. Supervised by Reed Solomon and Damir Dzhafarov. MSC: 03C57, 03B30.

Published online by Cambridge University Press:  15 December 2025

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Abstract

We say that a computable structure $\mathcal {A}$ is computably categorical if for every computable copy $\mathcal {B}$, there exists a computable isomorphism $f:\mathcal {A}\to \mathcal {B}$. This notion can be relativized to a degree $\mathbf {d}$ by saying that a computable structure $\mathcal {A}$ is computably categorical relative to $\mathbf {d}$ if for every $\mathbf {d}$-computable copy $\mathcal {B}$ of $\mathcal {A}$, there exists a $\mathbf {d}$-computable isomorphism $f:\mathcal {A}\to \mathcal {B}$. A key part of this thesis is to study the behavior of this notion of categoricity in the computably enumerable degrees.

The main theorem in Chapter $1$ states that given any computable partially ordered set P and any computable partition $P=P_0\sqcup P_1$, there exists an embedding h of P into the c.e. degrees and a computable graph $\mathcal {G}$ which is computably categorical, computably categorical relative to all degrees in $h(P_0)$, and is not computably categorical relative to any degree in $h(P_1)$. We also show that by using largely the same techniques alongside a standard construction of minimal pairs, we can embed a four-element diamond lattice into the c.e. degrees in the style of the main result of Chapter $1$.

We then apply some of the techniques used in this thesis to study the behavior of this notion in the context of generic degrees in Chapter $2$. Additionally, we show that several classes of structures admit a computable example that witnesses the pathological behavior of categoricity relative to a degree as seen in Chapter $1$’s main theorem.

Lastly, in the context of reverse mathematics, we investigate the reverse mathematical strength of a topological principle named $\mathsf {wGS}^{\operatorname {cl}}$, a weakened version of the Ginsburg–Sands theorem which states that every infinite topological space contains one of the following five topologies as a subspace, with $\mathbb {N}$ as the underlying set: discrete, indiscrete, cofinite, initial segment, or final segment.

Abstract prepared by Java Darleen Villano

E-mail: java.villano@utoronto.ca

URL: https://javavillano.github.io/thesisfinal.pdf

Information

Type
Thesis Abstract
Information
Bulletin of Symbolic Logic , Volume 31 , Issue 4 , December 2025 , pp. 694
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Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of The Association for Symbolic Logic

The Association for Symbolic Logic publishes abstracts of recent PhD theses in logic. The aim of this activity is to publish abstracts for the majority of recent PhD theses in logic world wide and submitted abstracts will therefore only be edited to ensure that they fall within the general area of logic and are appropriate in terms of length and content. This section willprovide a permanent publicly accessible overview of theses in logic and thus make up for the lack of central repository for the theses themselves. The Thesis Abstracts Section is edited by Sandra Müller. Any abstract should formally be submitted by the thesis advisor though it is expected to usually be prepared by the candidate. For detailed instructions for preparation and submission, including the required TeX template, please consult the link below. https://aslonline.org/journals/the-bulletin-of-symbolic-logic/logic-thesis-abstracts-in-the-bulletin-of-symbolic-logic/.