Kurt Gödel Research Center

2013 seminar talk: Cichon's diagram and regularity properties

Talk held by Yurii Khomskii (KGRC) at the KGRC seminar on 2013-03-21.

Abstract

The study of regularity properties in descriptive set theory is closely related to cardinal invariants of the continuum. Specifically, if we consider the Baire property, Lebesgue measure, and the Laver-, Miller- and Sacks-regularity for $\Sigma^1_2$ and $\Delta^1_2$ sets of reals, we obtain a pattern analogous to the well-known Cichon's diagram.

In this project we want to see what happens to this diagram on the $\Sigma^1_3$, $\Delta^1_3$ and higher projective levels. It is well-known that in the presence of a measurable cardinal, the picture lifts from the second to the third level, but without these assumptions, one can observe more interesting and surprising patterns. Consequently, all our results are proved on the basis of ZFC alone or ZFC with an inaccessible cardinal. An important technical ingredient in our proofs is the use of Suslin and Suslin+ proper (not necessarily ccc) forcing.

This is joint work with Vera Fischer and Sy Friedman.