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Solovay models and forcing extensions

  • Joan Bagaria (a1) and Roger Bosch (a2)

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- absolutely-ccc forcing extension, and that this is essentially the optimal preservation result, i.e., it does not hold for 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.

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Solovay models and forcing extensions

  • Joan Bagaria (a1) and Roger Bosch (a2)

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