Journal of Symbolic Logic

Solovay models and forcing extensions

Joan Bagaria and Roger Bosch

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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 δ31 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.

Article information

Source
J. Symbolic Logic, Volume 69, Issue 3 (2004), 742-766.

Dates
First available in Project Euclid: 4 October 2004

Permanent link to this document
https://projecteuclid.org/euclid.jsl/1096901764

Digital Object Identifier
doi:10.2178/jsl/1096901764

Mathematical Reviews number (MathSciNet)
MR2078919

Zentralblatt MATH identifier
1070.03031

Subjects
Primary: 03E15: Descriptive set theory [See also 28A05, 54H05] 03E35: Consistency and independence results

Keywords
Solovay models generic absoluteness definably-Mahlo cardinals productive-ccc partial orderings

Citation

Bagaria, Joan; Bosch, Roger. Solovay models and forcing extensions. J. Symbolic Logic 69 (2004), no. 3, 742--766. doi:10.2178/jsl/1096901764. https://projecteuclid.org/euclid.jsl/1096901764


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