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  1. 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.
    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).
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  • The stationarity of the collection of the locally regulars.Gunter Fuchs - 2015 - Archive for Mathematical Logic 54 (5-6):725-739.
    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.
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  • Generic Σ3 1 absoluteness.Sy D. Friedman - 2004 - Journal of Symbolic Logic 69 (1):73-80.
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  • Solovay models and forcing extensions.Joan Bagaria & Roger Bosch - 2004 - Journal of Symbolic Logic 69 (3):742-766.
    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 (...)
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  • Proper forcing extensions and Solovay models.Joan Bagaria & Roger Bosch - 2004 - Archive for Mathematical Logic 43 (6):739-750.
    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 (...)
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