Solovay models and forcing extensions

Journal of Symbolic Logic 69 (3):742-766 (2004)

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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DOI 10.2178/jsl/1096901764
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References found in this work BETA

Generic Absoluteness.Joan Bagaria & Sy D. Friedman - 2001 - Annals of Pure and Applied Logic 108 (1-3):3-13.
Some Applications of Almost Disjoint Forcing.R. B. Jensen & R. M. Solovay - 1970 - In Yehoshua Bar-Hillel (ed.), Mathematical Logic and Foundations of Set Theory. Amsterdam: North-Holland Pub. Co..
Projective Forcing.Joan Bagaria & Roger Bosch - 1997 - Annals of Pure and Applied Logic 86 (3):237-266.
Set Theory.Thomas Jech - 1999 - Studia Logica 63 (2):300-300.

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Citations of this work BETA

Forcing Absoluteness and Regularity Properties.Daisuke Ikegami - 2010 - Annals of Pure and Applied Logic 161 (7):879-894.
Parameterized Partition Relations on the Real Numbers.Joan Bagaria & Carlos A. Di Prisco - 2009 - Archive for Mathematical Logic 48 (2):201-226.
On Coding Uncountable Sets by Reals.Joan Bagaria & Vladimir Kanovei - 2010 - Mathematical Logic Quarterly 56 (4):409-424.
Absoluteness Via Resurrection.Giorgio Audrito & Matteo Viale - 2017 - Journal of Mathematical Logic 17 (2):1750005.
The Stationarity of the Collection of the Locally Regulars.Gunter Fuchs - 2015 - Archive for Mathematical Logic 54 (5-6):725-739.

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