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Some applications of almost disjoint forcing

In Yehoshua Bar-Hillel (ed.), Mathematical Logic and Foundations of Set Theory. Amsterdam: North-Holland Pub. Co. (1970)

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  1. L‐Mahlo Cardinals.Paul E. Cohen - 1974 - Mathematical Logic Quarterly 20 (13‐18):229-231.
  • L-Mahlo Cardinals.Paul E. Cohen - 1974 - Zeitschrift fur mathematische Logik und Grundlagen der Mathematik 20 (13-18):229-231.
  • The Modal Logic of -Centered Forcing and Related Forcing Classes.Ur Ya’Ar - 2021 - Journal of Symbolic Logic 86 (1):1-24.
    We consider the modality “ $\varphi $ is true in every $\sigma $ -centered forcing extension,” denoted $\square \varphi $, and its dual “ $\varphi $ is true in some $\sigma $ -centered forcing extension,” denoted $\lozenge \varphi $, which give rise to the notion of a principle of $\sigma $ -centered forcing. We prove that if ZFC is consistent, then the modal logic of $\sigma $ -centered forcing, i.e., the ZFC-provable principles of $\sigma $ -centered forcing, is exactly $\mathsf (...)
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  • Ns Saturated and -Definable.Stefan Hoffelner - 2021 - Journal of Symbolic Logic 86 (1):25-59.
    We show that under the assumption of the existence of the canonical inner model with one Woodin cardinal $M_1$, there is a model of $\mathsf {ZFC}$ in which $\mbox {NS}_{\omega _{1}}$ is $\aleph _2$ -saturated and ${\Delta }_{1}$ -definable with $\omega _1$ as a parameter which answers a question of S. D. Friedman and L. Wu. We also show that starting from an arbitrary universe with a Woodin cardinal, there is a model with $\mbox {NS}_{\omega _{1}}$ saturated and ${\Delta }_{1}$ (...)
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  • -Definability at Uncountable Regular Cardinals.Philipp Lücke - 2012 - Journal of Symbolic Logic 77 (3):1011-1046.
    Let k be an infinite cardinal. A subset of $(^k k)^n $ is a $\Sigma _1^1 $ -subset if it is the projection p[T] of all cofinal branches through a subtree T of $(lt;kk)^{n + 1} $ of height k. We define $\Sigma _k^1 - ,\Pi _k^1 $ - and $\Delta _k^1$ subsets of $(^k k)^n $ as usual. Given an uncountable regular cardinal k with k = k (...))
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  • Boolean Extensions Which Efface the Mahlo Property.William Boos - 1974 - Journal of Symbolic Logic 39 (2):254-268.
  • Forcing the Failure of Ch by Adding a Real.Saharon Shelah & Hugh Woodin - 1984 - Journal of Symbolic Logic 49 (4):1185-1189.
  • Combinatorics on Ideals and Forcing with Trees.Marcia J. Groszek - 1987 - Journal of Symbolic Logic 52 (3):582-593.
    Classes of forcings which add a real by forcing with branching conditions are examined, and conditions are found which guarantee that the generic real is of minimal degree over the ground model. An application is made to almost-disjoint coding via a real of minimal degree.
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  • Definability and Initial Segments of C-Degrees.Robert S. Lubarsky - 1988 - Journal of Symbolic Logic 53 (4):1070-1081.
    We combine two techniques of set theory relating to minimal degrees of constructibility. Jensen constructed a minimal real which is additionally a Π 1 2 singleton. Groszek built an initial segment of order type 1 + α * , for any ordinal α. This paper shows how to force a Π 1 2 singleton such that the c-degrees beneath it, all represented by reals, are of type 1 + α * , for many ordinals α. We also examine the definability (...)
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  • Proper Forcing and Remarkable Cardinals.Ralf-Dieter Schindler - 2000 - Bulletin of Symbolic Logic 6 (2):176-184.
  • Jensen's Σ* Theory and the Combinatorial Content of V = L.Sy D. Friedman - 1994 - Journal of Symbolic Logic 59 (3):1096 - 1104.
  • The Independence Of.Amir Leshem & Menachem Magidor - 1999 - Journal of Symbolic Logic 64 (1):350-362.
  • 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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  • A Guide to "Coding the Universe" by Beller, Jensen, Welch.Sy D. Friedman - 1985 - Journal of Symbolic Logic 50 (4):1002-1019.
  • Set Theoretic Naturalism.Penelope Maddy - 1996 - Journal of Symbolic Logic 61 (2):490-514.
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