Boolean ultrapowers, creatures, cardinal characteristics

The field of the project is set theory, in particular, set theory of the reals, cardinal characteristics and large continuum. Cantors result (from 1874) that the cardinality of the real line is strictly bigger than the cardinality of a countable infinite set, was the first theorem about cardinal characteristics of the continuum. This is a central result for, e.g., real analysis: We often study notions of smallness, such as Lebesgue measure zero or meager, with the property that countable sets are small (so if the real line was countable, these notions would not make sense). A cardinal characteristic (also called cardinal invariant), is, roughly speaking, the minimal cardinal number for which such a smallness property (which holds for all countable sets) fails. Cichons diagram summarizes some ZFC provable inequalities between cardinal characteristics related to the meager, null and sigma-compact ideals. Separating all ten possible cardinals in Cichons diagram was a long-standing open problem, recently solved by Goldstern, Kellner and Shelah, under the assumption of four strongly compact cardinals; before that, we gave a simpler construction for eight values (joint with Kellner and Tonti) and later another, more complicated, construction of the ten (joint with Kellner and Shelah). These results leave several questions open: Can we get rid of the large cardinal assumption? Can we get arbitrary values for the cardinals involved? Can we add other characteristics? Can get other orders? To this end, we plan to employ methods such as creature forcings, matrix iterations, ultrafilter and finitely additive measure limits, Boolean ultrapowers, and possibly template iterations.

Over a three-year period starting in October 2020, my research project explored two distinct directions. The first focused on automorphisms of generalized Boolean algebras and had as goal to show all automorphisms are trivial. Together with Saharon Shelah and Jakob Kellner, we achieved the first step of making them densely trivial. The second step, proving global triviality, remains a future research goal, due to the complexity. This partial result is substantial and we hypothesize that it can be combined with a generalized version or Random forcing to obtain the end result. In the second direction, we delved into Generalized Descriptive Set Theory, particularly studying Borel sets and their relations in a generalized space. Collaborating with Miguel Moreno, we considered topological and combinatorial aspects, highlighting the importance of understanding Borel* sets. A notable finding was the distinction between Borel and Borel* sets under the ideal topology, linking Generalized Descriptive Set Theory under singular cardinals with Generalized Descriptive Set Theory under different topologies. This connection opens a new research line with collaboration potential, with future applications is studying generalized cardinal characteristics of the continuum.