The seminar meets on Wednesday September 6th at 11:00 in the Institute

of Mathematics CAS, Zitna 25, seminar room, 3rd floor, front building.

Program:

Stefan Hoffelner — NS saturated and Δ_1-definable

Abstract:

Questions which investigate the interplay of the saturation of the

nonstationary ideal on ω_1, NS, and definability properties of the

surrounding universe can yield surprising and deep results. Woodins

theorem that in a model with a measurable cardinal where NS is

saturated, CH must definably fail is the paradigmatic example. It is

another remarkable theorem of H. Woodin that given ω-many Woodin

cardinals there is a model in which NS is saturated and ω-dense, which

in particular implies that NS is (boldface) Δ_1-definable. The latter

statement is of considerable interest in the emerging field of

generalized descriptive set theory, as the club filter is known to

violate the Baire property.

With that being said the following question, asked first by S.D.

Friedman and L. Wu seems relevant: is it possible to construct a model

in which NS is both Δ_1-definable and saturated from less than ω-many

Woodins? In this talk I will outline a proof that this is indeed the

case: given the existence of M_1^#, there is a model of ZFC in which the

nonstationary ideal on ω_1 is saturated and Δ_1-definable with parameter

ω_1. In the course of the proof I will present a new coding technique

which seems to be quite suitable to obtain definability results in the

presence of iterated forcing constructions over inner models for large

cardinals.