For a large natural class of forcing notions, we prove general equivalence theorems between forcing absoluteness statements, regularity properties, and transcendence properties over L and the core model K. We use our results to answer open questions from set theory of the reals.
We show that 1 3 -absoluteness for Sacks forcing is equivalent to the nonexistence of a 1 2 Bernstein set. We also show that Sacks forcing is the weakest forcing notion among all of the preorders that add a new real with respect to 1 3 forcing absoluteness.
We study classes of Borel subsets of the real line R such as levels of the Borel hierarchy and the class of sets that are reducible to the set Q of rationals, endowed with the Wadge quasi-order of reducibility with respect to continuous functions on R. Notably, we explore several structural properties of Borel subsets of R that diverge from those of Polish spaces with dimension zero. Our first main result is on the existence of embeddings of several posets into the restriction of this quasi-order to any Borel class that is strictly above the classes of open and closed sets, for instance the linear order ω 1 , its reverse ω ‹ 1 and the poset Ppωq{fin of inclusion modulo finite error. As a consequence of its proof, it is shown that there are no complete sets for these classes. We further extend the previous theorem to targets that are reducible to Q. These non-structure results motivate the study of further restrictions of the Wadge quasi-order. In our second main theorem, we introduce a combinatorial property that is shown to characterize those Fσ sets that are reducible to Q. This is applied to construct a minimal set below Q and prove its uniqueness up to Wadge equivalence. We finally prove several results concerning gaps and cardinal characteristics of the Wadge quasi-order and thereby answer questions of Brendle and Geschke.
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