Regularity Properties and Definability in the Real Number Continuum: idealized forcing, polarized partitions, Hausdorff gaps and mad families in the projective hierarchy. Yurii Khomskii Abstract: In this dissertation we study questions relevant to the foundations of mathematics, particularly the real number continuum. We look at regularity properties of sets of real numbers on one hand, and definability of such sets on the other. By ``regularity properties'' we are referring to certain desirable properties of sets of reals, something that makes such sets well-behaved, conforming to our intuition or easy to study. Classical examples include Lebesgue measurability, the property of Baire and the perfect set property. By ``definability'' we are referring to the possibility of giving an explicit description of a set. Classical examples of definable sets are the Borel, analytic and projective sets, and this leads to a measure of complexity of a set, whereby a set is considered as complex as the logical formula defining it. The relationship between regularity and definability has been known since the beginning of the 20th century. For example, all Borel and analytic sets satisfy most regularity properties. Using the Axiom of Choice, sets without such regularity properties can easily be constructed, but these are, in general, not definable. On the other hand, working in Gödel's constructible universe \$L\$, counterexamples can be found on the \$\SIGMA^1_2\$ level (the next level beyond the analytic). Typically, the assertion that all \$\SIGMA^1_2\$ or all \$\DELTA^1_2\$ sets of reals satisfy a certain regularity property is independent of ZFC, the standard axiomatization of set theory. Moreover, such an assertion can itself be seen as a possible additional hypothesis, implying among other things that the set-theoretic universe is larger than \$L\$ in a certain way. The focus of this dissertation is the interplay between regularity properties and definability, particularly the connection between regularity hypothesis and meta-mathematical statements about the set theoretic universe. In Chapter 2 we provide an abstract treatment of the above phenomenon, formulated in the framework of Idealized Forcing introduced by Jindřich Zapletal. We generalize some well-known theorems in this field and also a recent result of Daisuke Ikegami, and isolate several interesting questions. All proofs in this chapter rely heavily on the method of forcing. In Chapter 3 we consider the polarized partition property, a regularity property motivated by combinatorial questions and a relative of the classical Ramsey property. It has been studied in recent work of Carlos A. Di Prisco and Stevo Todorčević, among others. We prove several results relating this to other well-known regularity properties on the \$\SIGMA^1_2\$ and \$\DELTA^1_2\$ level. In Chapter 4 we turn our attention to Hausdroff gaps, classical objects known since the early 20th century which have numerous applications in various fields of mathematics such as topology and analysis. We extend a theorem of Stevo Todorčević stating that there are no analytic Hausdorff gaps in several directions. In Chapter 5 we look at maximal almost disjoint (mad) families from the definable point of view. We introduce a new notion of indestructibility of a mad family by forcing extensions, and using this notion prove a preservation result establishing the consistency of \$\bb > \aleph_1\$ together with the existence of a \$\SIGMA^1_2\$ definable mad family. This anwers a question posed by Sy Friedman and Lyubomyr Zdomskyy.