On a Fragment of the Universal Baire Property for Sigma^1_2 Sets

Abstract

There is a well-known global equivalence between \Sigma^1_2 sets having the Universal Baire property, two-step \Sigma^1_3 generic absoluteness, and the closure of the universe under the sharp operation. In this note, we determine the exact consistency strength of \Sigma^1_2 sets being (2^{\omega})^{+}-cc Universally Baire, which is below 0^{#}. In a model obtained, there is a \Sigma^1_2 set which is weakly \omega_2-Universally Baire but not \omega_2-Universally Baire.