Solovay models and forcing extensions

Journal of Symbolic Logic 69 (3):742-766 (2004)
Abstract
We study the preservation under projective ccc forcing extensions of the property of L(ℝ) being a Solovay model. We prove that this property is preserved by every strongly-̰Σ₃¹ absolutely-ccc forcing extension, and that this is essentially the optimal preservation result, i.e., it does not hold for Σ₃¹ absolutely-ccc forcing notions. We extend these results to the higher projective classes of ccc posets, and to the class of all projective ccc posets, using definably-Mahlo cardinals. As a consequence we obtain an exact equiconsistency result for generic absoluteness under projective absolutely-ccc forcing notions
Keywords Solovay models   generic absoluteness   definably-Mahlo cardinals   productive-ccc partial orderings
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Daisuke Ikegami (2010). Forcing Absoluteness and Regularity Properties. Annals of Pure and Applied Logic 161 (7):879-894.
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