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Proper forcing extensions and Solovay models

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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 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.

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Correspondence to Roger Bosch.

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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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Bagaria, J., Bosch, R. Proper forcing extensions and Solovay models. Arch. Math. Logic 43, 739–750 (2004). https://doi.org/10.1007/s00153-003-0210-2

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