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  1. A Co-Analytic Maximal Set of Orthogonal Measures.Vera Fischer & Asger Törnquist - 2010 - Journal of Symbolic Logic 75 (4):1403-1414.
    We prove that if V = L then there is a $\Pi _{1}^{1}$ maximal orthogonal (i.e., mutually singular) set of measures on Cantor space. This provides a natural counterpoint to the well-known theorem of Preiss and Rataj [16] that no analytic set of measures can be maximal orthogonal.
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  • Forcing Minimal Degree of Constructibility.Haim Judah & Saharon Shelah - 1991 - Journal of Symbolic Logic 56 (3):769-782.
    In this paper we will study four forcing notions, two of them giving a minimal degree of constructibility. These constructions give answers to questions in [Ih].
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  • 1996–1997 Winter Meeting of the Association for Symbolic Logic.Alexander S. Kechris - 1997 - Bulletin of Symbolic Logic 3 (3):367-377.
  • A Minimal Prikry-Type Forcing for Singularizing a Measurable Cardinal.Peter Koepke, Karen Räsch & Philipp Schlicht - 2013 - Journal of Symbolic Logic 78 (1):85-100.
    Recently, Gitik, Kanovei and the first author proved that for a classical Prikry forcing extension the family of the intermediate models can be parametrized by $\mathscr{P}(\omega)/\mathrm{finite}$. By modifying the standard Prikry tree forcing we define a Prikry-type forcing which also singularizes a measurable cardinal but which is minimal, i.e., there are \emph{no} intermediate models properly between the ground model and the generic extension. The proof relies on combining the rigidity of the tree structure with indiscernibility arguments resulting from the normality (...)
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  • Mathias Absoluteness and the Ramsey Property.Lorenz Halbeisen & Haim Judah - 1996 - Journal of Symbolic Logic 61 (1):177-194.
    In this article we give a forcing characterization for the Ramsey property of Σ 1 2 -sets of reals. This research was motivated by the well-known forcing characterizations for Lebesgue measurability and the Baire property of Σ 1 2 -sets of reals. Further we will show the relationship between higher degrees of forcing absoluteness and the Ramsey property of projective sets of reals.
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  • Amoeba-Absoluteness and Projective Measurability.Jörg Brendle - 1993 - Journal of Symbolic Logic 58 (4):1284-1290.
    We show that Σ1 4-Amoeba-absoluteness implies that $\forall a \in \mathbb{R}(\omega^{L\lbrack a \rbrack}_1 < \omega^V_1)$ and, hence, Σ1 3-measurability. This answers a question of Haim Judah (private communication).
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  • Ideals Without CCC.Marek Balcerzak, Andrzej RosŁanowski & Saharon Shelah - 1998 - Journal of Symbolic Logic 63 (1):128-148.
    Let I be an ideal of subsets of a Polish space X, containing all singletons and possessing a Borel basis. Assuming that I does not satisfy ccc, we consider the following conditions (B), (M) and (D). Condition (B) states that there is a disjoint family F $\subseteq$ P(X) of size c, consisting of Borel sets which are not in I. Condition (M) states that there is a Borel function f: X → X with $f^{-1}[\{x\}] \not\in$ I for each x ∈ (...)
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  • The Borel Conjecture.Haim Judah, Saharon Shelah & W. H. Woodin - 1990 - Annals of Pure and Applied Logic 50 (3):255-269.
    We show the Borel Conjecture is consistent with the continuum large.
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  • Combinatorial Properties of Classical Forcing Notions.Jörg Brendle - 1995 - Annals of Pure and Applied Logic 73 (2):143-170.
    We investigate the effect of adding a single real on cardinal invariants associated with the continuum. We show:1. adding an eventually different or a localization real adjoins a Luzin set of size continuum and a mad family of size ω1;2. Laver and Mathias forcing collapse the dominating number to ω1, and thus two Laver or Mathias reals added iteratively always force CH;3. Miller's rational perfect set forcing preserves the axiom MA.
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  • Regularity Properties for Dominating Projective Sets.Jörg Brendle, Greg Hjorth & Otmar Spinas - 1995 - Annals of Pure and Applied Logic 72 (3):291-307.
    We show that every dominating analytic set in the Baire space has a dominating closed subset. This improves a theorem of Spinas [15] saying that every dominating analytic set contains the branches of a uniform tree, i.e. a superperfect tree with the property that for every splitnode all the successor splitnodes have the same length. In [15], a subset of the Baire space is called u-regular if either it is not dominating or it contains the branches of a uniform tree, (...)
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  • Combinatorics and Forcing with Distributive Ideals.Pierre Matet - 1997 - Annals of Pure and Applied Logic 86 (2):137-201.
    We present a version for κ-distributive ideals over a regular infinite cardinal κ of some of the combinatorial results of Mathias on happy families. We also study an associated notion of forcing, which is a generalization of Mathias forcing and of Prikry forcing.
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  • Summable Gaps.James Hirschorn - 2003 - Annals of Pure and Applied Logic 120 (1-3):1-63.
    It is proved, under Martin's Axiom, that all gaps in are indestructible in any forcing extension by a separable measure algebra. This naturally leads to a new type of gap, a summable gap. The results of these investigations have applications in Descriptive Set Theory. For example, it is shown that under Martin's Axiom the Baire categoricity of all Δ31 non-Δ31-complete sets of reals requires a weakly compact cardinal.
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  • Construction with Opposition: Cardinal Invariants and Games.Jörg Brendle, Michael Hrušák & Víctor Torres-Pérez - forthcoming - Archive for Mathematical Logic:1-21.
    We consider several game versions of the cardinal invariants \, \ and \. We show that the standard proof that parametrized diamond principles prove that the cardinal invariants are small actually shows that their game counterparts are small. On the other hand we show that \ and \ are both relatively consistent with ZFC, where \ and \ are the principal game versions of \ and \, respectively. The corresponding question for \ remains open.
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  • Some Considerations on Amoeba Forcing Notions.Giorgio Laguzzi - 2014 - Archive for Mathematical Logic 53 (5-6):487-502.
  • Cichoń’s Diagram, Regularity Properties and $${\Varvec{\Delta}^1_3}$$ Δ 3 1 Sets of Reals.Vera Fischer, Sy David Friedman & Yurii Khomskii - 2014 - Archive for Mathematical Logic 53 (5-6):695-729.
  • Projective Absoluteness for Sacks Forcing.Daisuke Ikegami - 2009 - Archive for Mathematical Logic 48 (7):679-690.
    We show that ${{\bf \Sigma}^1_3}$ -absoluteness for Sacks forcing is equivalent to the non-existence of a ${{\bf \Delta}^1_2}$ Bernstein set. We also show that Sacks forcing is the weakest forcing notion among all of the preorders that add a new real with respect to ${{\bf \Sigma}^1_3}$ forcing absoluteness.
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  • Borel on the Questions Versus Borel on the Answers.Heike Mildenberger - 1999 - Mathematical Logic Quarterly 45 (1):127-133.
    We consider morphisms between binary relations that are used in the theory of cardinal characteristics. In [8] we have shown that there are pairs of relations with no Borel morphism connecting them. The reason was a strong impact of the first of the two functions that constitute a morphism, the so-called function on the questions. In this work we investigate whether the second half, the function on the answers' side, has a similarly strong impact. The main question is: Does the (...)
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  • Polarized Partitions on the Second Level of the Projective Hierarchy.Jörg Brendle & Yurii Khomskii - 2012 - Annals of Pure and Applied Logic 163 (9):1345-1357.
  • Forcing Absoluteness and Regularity Properties.Daisuke Ikegami - 2010 - Annals of Pure and Applied Logic 161 (7):879-894.
    For a large natural class of forcing notions, we prove general equivalence theorems between forcing absoluteness statements, regularity properties, and transcendence properties over and the core model . We use our results to answer open questions from set theory of the reals.
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  • On the Separation of Regularity Properties of the Reals.Giorgio Laguzzi - 2014 - Archive for Mathematical Logic 53 (7-8):731-747.
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  • On the Structure of Δ 1 4 -Sets of Reals.Haim Judah & Otmar Spinas - 1995 - Archive for Mathematical Logic 34 (5):301-312.
    Assuming that an inaccessible cardinal exists, we construct a ZFC-model where every Δ 1 4 -set is measurable but there exists a Δ 1 4 -set without the property of Baire. By a result of Shelah, an inaccessible cardinal is necessary for this result.
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  • Regularity Properties on the Generalized Reals.Sy David Friedman, Yurii Khomskii & Vadim Kulikov - 2016 - Annals of Pure and Applied Logic 167 (4):408-430.
  • Happy and Mad Families in L.Itay Neeman & Zach Norwood - 2018 - Journal of Symbolic Logic 83 (2):572-597.
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  • Sets of Reals.Joan Bagaria & W. Hugh Woodin - 1997 - Journal of Symbolic Logic 62 (4):1379-1428.
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  • -Stability.Dror Ben-Arié & Haim Judah - 1993 - Journal of Symbolic Logic 58 (3):941-954.
  • 1\ Sets of Reals.J. Bagaria & W. H. Woodin - 1997 - Journal of Symbolic Logic 62 (4):1379-1428.