Abstract:
The topic of this doctoral work is at the intersection of descriptive set theory, set theory
of the reals, and forcing.We study methods of obtaining models of ZFC plus the negation
of the continuumhypothesis inwhich there exist various combinatorially significant
subsets of the real line that are definable by a projective formula of provably minimal
complexity, i.e. is optimal.We present a general framework for obtaining the existence
of coanalytic combinatorial sets of reals given the existence of aΣ12
such witness. This is
done by collecting a series of these reduction theorems appearing sporadically throughout
the literature, extracting fromtheir proofs an overarching pattern, and presenting
them in a uniform fashion. These theorems improve previous constructions which required
the assumption V = L. The general strategy is applied to provide a new reduction
theorem for the case of Hausdorff gaps.We then study forcing notions relevant to
the theory of cardinal characteristics; first we show a proper forcing given by Shelah
in 1984 and its countable support iterations preserve tight mad families, providing an
alternative proof of the consistency of b = a < s. This is applied to show that a < s is
consistent with aΔ13
wellordering of the reals and a coanalytic tight mad family witnessing
a = ℵ1, responding to questions of Fischer and Friedman.We extend the theory of
projective witnesses for values of cardinal characteristics by considering the spectrum
of a, the set of possible cardinalities of a mad family. By showing a forcing notion of
Friedman and Zdomskyy also preserves tight mad families in a strong sense, we obtain
the consistency of a = ℵ1 < c = ℵ2, and for every κ in the spectrum of a there exists a
projective tight mad family of size κ with definition of optimal complexity. In the end
we show the compatibility of our results and construct a model witnessing all of the
conclusions simultaneously.

Online:
univienna.zoom.us/j/62138828442
Meeting-ID: 621 3882 8442
Kenncode: 668427