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Generic absoluteness

Joan Bagaria & Sy D. Friedman

Annals of Pure and Applied Logic 108 (1-3):3-13 (2001)

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Proper forcing extensions and Solovay models.Joan Bagaria & Roger Bosch - 2004 - Archive for Mathematical Logic 43 (6):739-750.details 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 (...) Download Export citation Bookmark 6 citations
Solovay models and forcing extensions.Joan Bagaria & Roger Bosch - 2004 - Journal of Symbolic Logic 69 (3):742-766.details 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 Σ₃¹ 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 (...) Download Export citation Bookmark 6 citations
The Necessary Maximality Principle for c. c. c. forcing is equiconsistent with a weakly compact cardinal.Joel D. Hamkins & W. Hugh Woodin - 2005 - Mathematical Logic Quarterly 51 (5):493-498.details The Necessary Maximality Principle for c. c. c. forcing with real parameters is equiconsistent with the existence of a weakly compact cardinal. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim). Download Export citation Bookmark 8 citations
The stationarity of the collection of the locally regulars.Gunter Fuchs - 2015 - Archive for Mathematical Logic 54 (5-6):725-739.details I analyze various natural assumptions which imply that the set {ω1L[x]∣x⊆ω}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\{\omega_1^{L[x]} \mid x \subseteq \omega\}}$$\end{document} is stationary in ω1. The focal questions are which implications hold between them, what their consistency strengths are, and which large cardinal assumptions outright imply them. Download Export citation Bookmark
Generic Σ3 1 absoluteness.Sy D. Friedman - 2004 - Journal of Symbolic Logic 69 (1):73-80.details Download Export citation Bookmark

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