Proper forcing extensions and Solovay models |
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| Authors: | Joan?Bagaria Email author" target="_blank">Roger?BoschEmail author |
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| Institution: | (1) Institució Cataloniaitució Catalana de Recerca i Estudis Avançats (ICREA) and Departament de Lògica, Història i Filosofia de la Ciència, Universitat de Barcelona, Baldiri Reixac, s/n, 08028 Barcelona, Catalonia, Spain;(2) Departmento de Filosofía, Universidad de Oviedo, Teniente Alfonso Martínez, s/n, 33011 Oviedo, Asturias, Spain |
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| Abstract: | We study the preservation of the property of  being a Solovay model under proper projective forcing extensions. We show that every  strongly-proper forcing notion preserves this property. This yields that the consistency strength of the absoluteness of under strongly-proper forcing notions is that of the existence of an inaccessible cardinal. Further, the absoluteness of under projective strongly-proper forcing notions is consistent relative to the existence of a  -Mahlo cardinal. We also show that the consistency strength of the absoluteness of under forcing extensions with  -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.Research partially supported by the research projects BFM2002-03236 of the Spanish Ministry of Science and Technology, and 2002SGR 00126 of the Generalitat de Catalunya. The second author was also partially supported by the research project GE01/HUM10, Grupos de excelencia, Principado de Asturias.Mathematics Subject Classification (2000): 03E15, 03E35 |
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| Keywords: | Solovay models Generic absoluteness Strongly-proper forcing Projective forcing Consistency strength Definably-Mahlo cardinals Mahlo cardinals Weakly-compact cardinals |
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