Generic absoluteness under projective forcing
Volume 194 / 2007
Fundamenta Mathematicae 194 (2007), 95-120
MSC: 03E15, 03E35, 03E50, 03E55.
DOI: 10.4064/fm194-2-1
Abstract
We study the preservation of the property of $\bf LR$ being a Solovay model under projective ccc forcing extensions. We compute the exact consistency strength of the generic absoluteness of $\bf LR$ under forcing with projective ccc partial orderings and, as an application, we build models in which Martin's Axiom holds for ${\mathop{\Sigma}\limits_{\textstyle\sim}}{}^1_n$ partial orderings, but it fails for the ${\mathop{\Sigma}\limits_{\textstyle\sim}}{}^1_{n+1}$.
