Projective absoluteness for Sacks forcing
This paper examines the connections between the absoluteness of projective sentences for the Sacks forcing extensions and regularity properties of \({\pmb\Sigma}^1_2\) and \({\pmb\Delta}^1_2\) sets of reals. The main result establishes the equivalence of \({\pmb\Sigma}^1_3\)-absoluteness for the Sacks forcing and non-existence of \({\pmb\Delta}^1_2\) Bernstein sets. The latter is known (by [\textit{J. Brendle} and \textit{B. Löwe}, ``Solovay-type characterizations for forcing-algebras'', J. Symb. Log. 64, No. 3, 1307--1323 (1999; Zbl 0945.03071)]) to be equivalent to non-existence of \({\pmb\Sigma}^1_2\) Bernstein sets, as well as to the fact that for each \(r\in\mathbb{R}\) there is a real which does not belong to \(L[r]\). The \({\pmb\Sigma}^1_3\)-absoluteness for the Sacks forcing is shown to be the weakest statement of this kind, i.e., for any forcing notion \(\mathbb{P}\) adding reals, \({\pmb\Sigma}^1_3\)-absoluteness for \(\mathbb{P}\) implies \({\pmb\Sigma}^1_3\)-absoluteness for the Sacks forcing. Also, the absoluteness of higher projective sentences is examined for the Sacks forcing extensions. It is proved that for each \(n\geq 1\), if \({\pmb\Sigma}^1_{n+1}\)-absoluteness holds for the Sacks forcing, then there are no \({\pmb\Delta}^1_n\) Bernstein sets. Under the assumption that the class \({\pmb\Pi}^1_{2n-1}\) has the uniformization property, it is shown that non-existence of \({\pmb\Delta}^1_{2n}\) Bernstein sets implies \({\pmb\Sigma}^1_{2n+1}\)-absoluteness for the Sacks forcing.
scientific article; zbMATH DE number 5620101