Some Contributions to Generalised Descriptive Set Theory and to the Study of the Interplay between Modal Logic and Set Theory.

This dissertation is divided into two main parts. The first part, which brings together two separate projects, contributes to the field of generalised descriptive set theory (GDST). The second part explores the relationship between modal logic and set theory. Chapter 1 is joint work with Vincenzo Dimonte and Sandra Müller. Chapter 2 is the result of joint research with Claudio Agostini and Vincenzo Dimonte. Finally, Chapter 3 presents work developed in collaboration with Juan P. Aguilera and Grigorii Stepanov. The first part of this work focuses primarily on the topic of (generalised) regularity properties. In particular, in Chapter 1 we prove that, given a strong limit cardinal \lambda of countable cofinality, if every \lambda-coanalytic subset of the generalized Cantor space {}^{\lambda}2 has the \lambda-PSP, a straigthforward generalisation of the Perfect Set Property, then 0^\dagger exists. To prove this, we raise this lower bound, starting from the existence of an inner model with a single measurable cardinal and building up to the mentioned stronger assumption. Our analysis makes use of the inner model theory and the covering properties of the Dodd-Jensen core model and L[U]. Perhaps the main contribution of this work to the field of generalised descriptive set theory is not the specific mentioned result, but rather the analysis of sets of codes in {}^{\lambda}2 for mice in the Dodd-Jensen model and L[U], along with a study of their (generalised) descriptive complexity. In Chapter 2, we investigate whether a generalised notion of measure can exist in the generalised Baire space, ultimately arriving at a negative conclusion: by assuming very mild assumptions and adopting an "umbrella definition" that encompasses all possible natural generalisations of classical measures, we conclude that no non-trivial and continuous generalised measure satisfying reasonable structural axioms can exist. In Chapter 3, we study the interplay between modal logic and set theory. Specifically, we consider two modal operators, \Box and \bBox, interpreted respectively as "it holds in every forcing extension" and "it holds in every ground". We propose a modal logic to characterise this interpretation, characterise it in terms of Kripke frames and prove that it is indeed sound and complete with respect to these semantics. We also briefly discuss steps towards developing a similar framework where \bBox is instead interpreted as "it holds in every inner model".