Stefan Hoffelner: NS saturated and Δ_1-definable

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.

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