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.