Forcing Absoluteness and Regularity Properties

(Abstract)

There is a close connection between forcing absolutenss and regularity properties, e.g., Sigma^1_3 Cohen forcing absoluteness holds if and only if every Delta^1_2 set of reals has the Baire property. It is known that the same kind of phenomena occur for random forcing and Lebesgue measurability, Mathias forcing and completely Ramseyness, Sacks forcing and Bernstein property etc. In this talk, I will introduce a class of forcing notions from which we can define the corresponding regularity properties and prove the equivalence as above for each forcing in the class in a uniform way. This class contains all the practical forcing notions connected to regularity properties and this result implies the unknown equivalence for some forcings (e.g. Silver forcing and Miller forcing). I will also talk about the connection between the regularity properties I define and the P-Baireness for a forcing P, which is essentially introduced in the paper "Universally Baire Sets of Reals" by Feng, Magidor and Woodin. If time permits, I will discuss some open problems in this topic.