Stefan Hoffelner: NS saturated and Delta_1-definable

Monday, June 19, 2017, 16.30
Seminar room 0.011, Mathematical Institute, University of Bonn

Speaker: Stefan Hoffelner (University of Vienna)

Title: NS saturated and Delta_1-definable


Questions which investigate the interplay of the saturation of the nonstationary ideal on $omega_1$, 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 $omega$-many Woodin cardinals there is a model in which NS is saturated and $omega_1$-dense, which in particular implies that NS is (boldface) $Delta_1$-definable. S.D. Friedman and L. Wu asked whether the large cardinal assumption can be lowered while keeping NS $Delta_1$-definable and saturated. 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 $omega_1$ is saturated and $Delta_1$-definable with parameter $K_{omega_2^K}$ (note that $omega_2^K$ is of size $aleph_1$ in that model). 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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