Proper forcing extensions and Solovay models
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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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