I am invited to give a talk in the Sektion Logik at the DMV-ÖMG Annual Conference 2021 taking place virtually from Passau, Germany, Sep 27 - Oct 1, 2021.
Uniformization and internal absoluteness
Measurability with respect to ideals is known to be tightly connected to absoluteness principles for certain forcing notions. In joint work with Philipp Schlicht we study a uniformization principle that postulates the existence of a uniformizing function on a large set, relative to a given ideal. We prove that for all ideals I such that Borel/I is proper, this uniformization principle is equivalent to an absoluteness principle for projective formulas that we call internal absoluteness. In addition, we show that it is equivalent to measurability with respect to I together with 1-step absoluteness for the poset Borel/I. These equivalences are new even for Cohen and random forcing and are, to the best of our knowledge, the first precise equivalences between regularity and absoluteness beyond the second level of the projective hierarchy.