Infinitary Combinatorics and Definability

The combinatorial properties of infinite sets of reals have been of interest since the early development of analysis in the 19th century. It can even be said that the work of Georg Cantor from 1872 on the trigonometric series representation of a function marks the very beginning of the subject of Set Theory. Two ground-breaking developments in this subject are the appearance of Kurt Gödel`s Constructible Universe and Paul Cohen`s method of Forcing. The current project focusses on some of the central problems of the infinitary combinatorial properties of the real line, arising from the above two phenomenon, definability and independence.

The axiomatic system of Zermelo-Fraenkel, together with the method of forcing capture a richness of infinitary combinatorial structures in terms of special sets of reals, sets which often originate in analysis, topology or algebra. These special sets are often associated with the so called combinatorial cardinal characteristics of the real line, and can be defined in terms of elementary set theoretic operations. However, the infinitary combinatorics captured by them is extraordinarily complex and easily bring us to the boundaries of mathematics, as indicated by the abundance of independence and relative consistency results in the area. The infinitary combinatorics captured by these sets of reals can be seen as a common language between different, seemingly unrelated, mathematical objects, often objects belonging to seemingly unrelated areas of mathematics. Apart from establishing remarkable new relative consistency results between classical combinatorial cardinal characteristics, and so in particular between such sets of interest, we establish also several significant new ZFC dependencies. Two of the strongest results of the project are, on one side, the fact that consistently there is a maximal independent family with cardinality of countable cofinality, and on the other the fact that the minimal cardinality of an ultrafilter base never exceeds the minimal cardinality of a maximal ideal independent family. A cardinal characteristic can be seen as the least cardinality of a set of reals with a specific property, while the spectrum of such a characteristic is the set of all possible cardinalities of sets of reals satisfying a relevant property. The project makes significant advancements in understanding the spectrum of several cardinal characteristics. Controlling their spectra gives rise to a deep analysis of notions of strong maximality, providing these sets an additional combinatorial strength and so allowing us to establish the relative consistency of numerous, prior unknown, cardinal characteristic constellations. A parallel, and equally exciting theme of investigation is the study of the definability properties of such combinatorial sets of reals. The project makes breakthrough advancements in understanding the projective complexity of various combinatorial sets of reals in models with large continuum and provides models in which numerous constellations, witnessed by families of optimal projective complexity. Combining the study of the spectrum, together with the existence of witnesses of optimal projective complexity, brings us to the study of the so called models of optimal projective spectra: these are models in which each cardinal appearing in the spectrum of a given characteristic is witnessed by a relevant combinatorial set of optimal projective complexity. The work and results on the project form significant advancements in the study of the set theoretic properties of the real line, answering a large number of questions and bringing to light new exciting questions and directions of study.