Solovay models and forcing extensions. (English) Zbl 1070.03031

Bagaria, Joan; Bosch, Roger

J. Symb. Log. 69, No. 3, 742-766 (2004).

Summary: We study the preservation under projective ccc forcing extensions of the property of \(L(\mathbb R)\) being a Solovay model. We prove that this property is preserved by every strongly-\({\pmb\Sigma}^1_3\) absolutely-ccc forcing extension, and that this is essentially the optimal preservation result, i.e. it does not hold for \(\Delta^1_3\) 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.

MSC:

03E15	Descriptive set theory
03E35	Consistency and independence results

Keywords:

preservation under projective ccc forcing extensions; Solovay model; definably-Mahlo cardinals; equiconsistency; generic absoluteness

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