Algebra Seminar talk
2024-12-06
Lukas Koschat
Towards canonical sets of sets of reals and full models of LSA in L(uB)
Abstract:
In unpublished work, Müller and Sargsyan introduced a family of sets of reals, A∞, which consistently satisfies a certain forcing absoluteness, generalising the forcing absoluteness associated with universally Baire (uB) sets of reals. The talk will begin with an accessible overview, introducing the necessary background before defining A∞ and explaining the key result by Müller-Sargsyan on the associated forcing absoluteness.
Building on this, we present recent joint work with Müller and Sargsyan, where we compute the cofinality of the ordinal height of A∞ as ω under the assumption that the universe arises from collapsing a supercompact cardinal. This result serves as a foundational step toward identifying a maximal collection of canonical subsets of uB.
In the final part of the talk, I will discuss ongoing work aimed at constructing full models of the Largest Suslin Axiom (LSA) within the minimal proper class inner model containing all uB sets, where the background universe again arises from collapsing a supercompact cardinal. The connection between this work and the results on A∞ lies in their shared setup and the similarity of the underlying techniques.
There will be a second, more technical continuation talk in the afternoon (14:00-15:30) aimed at an expert audience.