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There are known characterisations of several fragments of hybrid logic by means of invariance under bisimulations of some kind. The fragments include 
$\{\mathord {\downarrow }, \mathord {@}\}$ with or without nominals (Areces, Blackburn, Marx), 
$\mathord {@}$ with or without nominals (ten Cate), and 
$\mathord {\downarrow }$ without nominals (Hodkinson, Tahiri). Some pairs of these characterisations, however, are incompatible with one another. For other fragments of hybrid logic no such characterisations were known so far. We prove a generic bisimulation characterisation theorem for all standard fragments of hybrid logic, in particular for the case with 
$\mathord {\downarrow }$ and nominals, left open by Hodkinson and Tahiri. Our characterisation is built on a common base and for each feature extension adds a specific condition, so it is modular in an engineering sense.
This article offers a philosophical overview and investigation of the problem of incompleteness in set theory and what this entails for the ensuing debates about proposed extensions of 
$ZFC$. The incompleteness of 
$ZFC$ is well-known and leaves us with a rich array of competing extensions. What should we make of disagreements between them? We start by considering second-order logic and its categoricity theorems and how they might be used to compare different set theories. We then aim to use interpretability as a way of understanding that some of these debates are insubstantial. This culminates in some discussion of the relationship between interpretability and the generic multiverse. The second half of the article then takes up a more modest goal: we search for common ground and settle for partial agreement between set theories in much the same way that physicists are often content with empirical agreement. We then aim to describe a natural bound on the amount of agreement that we can expect to obtain between reasonable extensions of 
$ZFC$.
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
This thesis presents my contributions to various aspects of the theory of universally Baire sets. One of these aspects is the smallest inner model containing all reals whose all sets of reals are universally Baire (viz., 
$L(\mathbb {R})$) and its relation to its inner model 
$\mathsf {HOD}$. We verify here that 
$\mathsf {HOD}^{L(\mathbb {R})}$ enjoys a form of local definability inside 
$L(\mathbb {R})$, further justifying its characterization as a “core model” in 
$L(\mathbb {R})$. We then study a “bottom-up” construction of more complicated universally Baire sets (more generally, determined sets). This construction allows us to give an “L-like” description of the minimum model of 
$\mathsf {AD}_{\mathbb {R}} + \mathsf {Cof}(\Theta ) = \Theta $. A consequence of this description is that this minimum model is contained in the Chang-plus model. Our construction, together with Woodin’s work on the Chang-plus model, shows that a proper class of Woodin cardinals which are limits of Woodin cardinals implies the existence of a hod mouse with a measurable limit of Woodin cardinals whose strategy is universally Baire.
Another aspect of the theory of universally Baire sets is the generic absoluteness and maximality associated with them. We include some results concerning generic 
$\Sigma _1^{H(\omega _2)}$-absoluteness with universally Baire sets as predicates or parameters, as well as generic 
$\Pi _2^{H(\omega _2)}$-maximality with universally Baire sets as predicates. In the second case, we are led to consider the general question of when a model of an infinitary propositional formula can be added by a stationary-set-preserving poset. We characterize when this happens in terms of a game which is a variant of the Model Existence Game. We then give a sufficient condition for this in terms of generic embeddings.
Abstract taken directly from the thesis
E-mail: obrad@math.ucla.edu