On a Fragment of the Universal Baire Property for Sigma¹₂ Sets

الأصل · EN

There is a well-known global equivalence between Σ¹₂ sets having the Universal Baire property, two-step Σ¹₃ generic absoluteness, and the closure of the universe under the sharp operation. In this note, we determine the exact consistency strength of Σ¹₂ sets being (2ω)⁺-cc Universally Baire, which is below 0#. In a model obtained, there is a Σ¹₂ set which is weakly ω₂-Universally Baire but not ω₂-Universally Baire.