Abstract. It is known that all the Σ13-statements (with real parameters) are absolute between the ground model and its Cohen forcing extensions iff every Δ12-set of reals has the property of Baire iff for any real x, there is a Cohen real over L[x]. This kind of equivalence holds for many forcings related to the reals. In this talk, I will introduce a large class of forcing notions containg almost all the typical forcings related to the reals and introduce the regularity property and prove the above kind of equivalence in a uniform way. I will also explore the connection between Σ14-absoluteness, the regularity property for Δ13-sets of reals and the transcendence property for the core model K.
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