Kurt Gödel Research Center

2012 seminar talk: Regularity Idealized

Talk held by Yurii Khomskii (KGRC) at the KGRC seminar on 2012-03-08.

Abstract

I would like to talk about regularity properties of sets of reals and how these can be uniformly defined within the abstract framework of "idealized forcing" developed by Jindrich Zapletal. All the classical regularity properties, such as Lebesgue measurability, the Baire, Ramsey and perfect set properties, as well as more recent ones such as Sacks-, Miller- and Laver-measurability, fit naturally into this framework. Consequently, many results regarding such properties, particularly in relation to descriptive set theory, can now be given uniform proofs.

This was one of the lines of research I pursued in my PhD thesis. Granted, many results presented here are strictly speaking not new but abstractions or generalizations of previously known results, such as those of Solovay, Judah-Shelah, Brendle, Loewe, Zapletal and Ikegami. However, the current framework seeks to explore and better explain the relationships between all these individual results. Moreover, many interesting open questions naturally arise along the way, and I would like to use this opportunity to present them to a larger audience.