2021/08/22 by Sandra Müller, Philipp Schlicht, Müller, Sandra +1
Computer Science · Mathematics · #Advanced Algebra and Logic #Advanced Topology and Set Theory #FOS: Mathematics #Logic (math.LO) #Rings, Modules, and Algebras
paper · doi:10.48550/arxiv.2108.09688
openalex publication_date 2021/08/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01
Measurability with respect to ideals is tightly connected with absoluteness principles for certain forcing notions. 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 σ-ideals I such that the ideal forcing ℙI of Borel sets modulo I is proper, this uniformization principle is equivalent to an absoluteness principle for projective formulas with respect to ℙI 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 ℙI. These equivalences are new even for Cohen and random forcing and they are, to the best of our knowledge, the first precise equivalences between regularity and absoluteness beyond the second level of the projective hierarchy.