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Archive for Mathematical Logic

, Volume 43, Issue 6, pp 739–750 | Cite as

Proper forcing extensions and Solovay models

  • Joan Bagaria
  • Roger BoschEmail author
Article

Abstract.

We study the preservation of the property of Open image in new window being a Solovay model under proper projective forcing extensions. We show that every Open image in new window strongly-proper forcing notion preserves this property. This yields that the consistency strength of the absoluteness of Open image in new window under Open image in new window strongly-proper forcing notions is that of the existence of an inaccessible cardinal. Further, the absoluteness of Open image in new window under projective strongly-proper forcing notions is consistent relative to the existence of a Open image in new window -Mahlo cardinal. We also show that the consistency strength of the absoluteness of Open image in new window under forcing extensions with σ-linked forcing notions is exactly that of the existence of a Mahlo cardinal, in contrast with the general ccc case, which requires a weakly-compact cardinal.

Key words or phrases:

Solovay models Generic absoluteness Strongly-proper forcing Projective forcing Consistency strength Definably-Mahlo cardinals Mahlo cardinals Weakly-compact cardinals 

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Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  1. 1.Institució Cataloniaitució Catalana de Recerca i Estudis Avançats (ICREA) and Departament de Lògica, Història i Filosofia de la CiènciaUniversitat de BarcelonaBarcelonaSpain
  2. 2.Departmento de FilosofíaUniversidad de OviedoOviedoSpain

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