Journal of Symbolic Logic

Solovay models and forcing extensions

Joan Bagaria and Roger Bosch

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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-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

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

First available in Project Euclid: 4 October 2004

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Zentralblatt MATH identifier

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

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


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

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  • J. Bagaria Definable forcing and regularity properties of projective sets of reals, Ph.D. thesis, University of California at Berkeley,1991.
  • J. Bagaria and R. Bosch Projective forcing, Annals of Pure and Applied Logic, vol. 86 (1997), pp. 237--266.
  • J. Bagaria and S. Friedman Generic absoluteness, Annals of Pure and Applied Logic, vol. 108 (2001), pp. 3--13.
  • J. Bagaria and H. Judah Amoeba forcing, Suslin absoluteness and additivity of measure, Set theory of the continuum (H. Judah, W. Just, and W. H. Woodin, editors), MSRI, Berkeley, Springer-Verlag, Berlin,1992.
  • T. Jech Set Theory, Springer Monographs in Mathematics, Springer-Verlag, Berlin, Heidelberg,2003, The third millenium edition, revised and expanded.
  • R. B. Jensen and R. M. Solovay Some applications of almost disjoint sets, Mathematical Logic and Foundations of Set Theory (Y. Bar-Hillel, editor), North Holland, Amsterdam,1970.
  • H. Judah and A. Rosłanowski Martin Axiom and the size of the continuum, Journal of Symbolic Logic, vol. 60 (1995), pp. 374--391.
  • H. Judah and S. Shelah Souslin forcing, Journal of Symbolic Logic, vol. 53 (1988), pp. 1188--1207.
  • A. Kanamori The higher infinite: Large cardinals in set theory from their beginnings, Perspectives in Mathematical Logic, Springer-Verlag, Berlin,1994.
  • S. Shelah and W. H. Woodin Large cardinals imply that every reasonably definable set of reals is Lebesgue measurable, Israel Journal of Mathematics, vol. 70 (1990), no. 3, pp. 381--394.
  • R. Solovay A model of set theory in which every set of reals is Lebesgue measurable, Annals of Mathematics, vol. 92 (1970), pp. 1--56.
  • W. H. Woodin On the consistency strength of projective uniformization, Logic Colloquium'81 (J. Stern, editor), North-Holland, Amsterdam,1982.