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Solovay models and forcing extensions

Published online by Cambridge University Press:  12 March 2014

Joan Bagaria and
Roger Bosch
Joan Bagaria
Affiliation:
Institució Catalana de Recerca i Estudis Avançats (Icrea), and Departament de Lògica, Història i Filosofia de la Ciència, Universitat de Barcelona, 08028 Barcelona, Catalonia, Spain, E-mail: bagaria@ub.edu
Roger Bosch
Affiliation:
Departamento de Filosofía, Universidad de Oviedo, 33071 Oviedo, Spain, E-mail: roger@pinon.ccu.uniovi.es
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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- 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.

Keywords

Solovay modelsgeneric absolutenessdefinably-Mahlo cardinalsproductive-ccc partial orderings

Information

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 69 , Issue 3 , September 2004, pp. 742 - 766
Copyright
Copyright © Association for Symbolic Logic 2004

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References

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