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Seminar on Reckoning

NS saturated and Δ1-definable

Stefan Hoffelner


Seminar on Reckoning
Wednesday, 6. September 2017 - 11:00 to 15:00

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

Questions which investigate the interplay of the saturation of the nonstationary ideal on ω1, 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) Δ1-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 Δ1-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 M1^#, there is a model of ZFC in which the nonstationary ideal on ω1 is saturated and Δ1-definable with parameter ω1. 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.