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Solovay models and forcing extensions
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- Journal:
- The Journal of Symbolic Logic / Volume 69 / Issue 3 / September 2004
- Published online by Cambridge University Press:
- 12 March 2014, pp. 742-766
- Print publication:
- September 2004
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- Article
- Export citation
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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.
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.