Generic Σ 1 3 Absoluteness
نویسنده
چکیده
In this article we study the strength of Σ 1 3 absoluteness (with real parameters) in various types of generic extensions, correcting and improving some results from [2]. We shall also make some comments relating this work to the bounded forcing axioms BMM, BPFA and BSPFA. The statement " Σ 1 3 absoluteness holds for ccc forcing " means that if a Σ 1 3 formula with real parameters has a solution in a ccc set-forcing extension of the universe V , then it already has a solution in V. The analogous definition applies when ccc is replaced by other set-forcing notions, or by class-forcing.
منابع مشابه
Generic absoluteness under projective forcing∗
We study the preservation of the property of L(R) being a Solovay model under projective ccc forcing extensions. We compute the exact consistency strength of the generic absoluteness of L(R) under forcing with projective ccc partial orderings and, as an application, we build models in which Martin’s Axiom holds for Σ ∼ 1 n partial orderings, but it fails for the Σ ∼ 1 n+1.
متن کاملUniversally Baire Sets and Generic Absoluteness
We prove several equivalences and relative consistency results involving notions of generic absoluteness beyond Woodin’s (Σ ̃ 1)λ generic absoluteness for a limit of Woodin cardinals λ. In particular, we prove that two-step ∃R(Π ̃ 1)λ generic absoluteness below a measurable cardinal that is a limit of Woodin cardinals has high consistency strength, and that it is equivalent with the existence of ...
متن کاملGeneric absoluteness
We explore the consistency strength of 1 3 and 1 4 absoluteness, for a variety of forcing notions.
متن کاملProper forcing extensions and Solovay models
We study the preservation of the property of L(R) being a Solovay model under proper projective forcing extensions. We show that every ∼ 1 3 strongly-proper forcing notion preserves this property. This yields that the consistency strength of the absoluteness of L(R) under ∼ 1 3 strongly-proper forcing notions is that of the existence of an inaccessible cardinal. Further, the absoluteness of L(R...
متن کامل