20 found
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  1. Generic Trees.Otmar Spinas - 1995 - Journal of Symbolic Logic 60 (3):705-726.
    We continue the investigation of the Laver ideal ℓ 0 and Miller ideal m 0 started in [GJSp] and [GRShSp]; these are the ideals on the Baire space associated with Laver forcing and Miller forcing. We solve several open problems from these papers. The main result is the construction of models for $t , where add denotes the additivity coefficient of an ideal. For this we construct amoeba forcings for these forcings which do not add Cohen reals. We show that (...)
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  2.  19
    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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  3. Analytic Countably Splitting Families.Otmar Spinas - 2004 - Journal of Symbolic Logic 69 (1):101-117.
    A family A ⊆ ℘(ω) is called countably splitting if for every countable $F \subseteq [\omega]^{\omega}$ , some element of A splits every member of F. We define a notion of a splitting tree, by means of which we prove that every analytic countably splitting family contains a closed countably splitting family. An application of this notion solves a problem of Blass. On the other hand we show that there exists an $F_{\sigma}$ splitting family that does not contain a closed (...)
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  4.  10
    Dominating Projective Sets in the Baire Space.Otmar Spinas - 1994 - Annals of Pure and Applied Logic 68 (3):327-342.
    We show that every analytic set in the Baire space which is dominating 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. We call this property of analytic sets u-regularity. However, we show that the concept of uniform tree does not suffice to characterize dominating analytic sets in general. We construct a dominating closed set with the property that for no uniform tree whose (...)
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  5.  5
    Silver antichains.Otmar Spinas & Marek Wyszkowski - 2015 - Journal of Symbolic Logic 80 (2):503-519.
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  6.  25
    Independence and Consistency Proofs in Quadratic Form Theory.James E. Baumgartner & Otmar Spinas - 1991 - Journal of Symbolic Logic 56 (4):1195-1211.
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  7.  11
    Partition Numbers.Otmar Spinas - 1997 - Annals of Pure and Applied Logic 90 (1-3):243-262.
    We continue [21] and study partition numbers of partial orderings which are related to /fin. In particular, we investigate Pf, be the suborder of /fin)ω containing only filtered elements, the Mathias partial order M, and , ω the lattice of partitions of ω, respectively. We show that Solomon's inequality holds for M and that it consistently fails for Pf. We show that the partition number of is C. We also show that consistently the distributivity number of ω is smaller than (...)
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  8.  12
    Dominating and Unbounded Free Sets.Slawomir Solecki & Otmar Spinas - 1999 - Journal of Symbolic Logic 64 (1):75-80.
    We prove that every analytic set in ω ω × ω ω with σ-bounded sections has a not σ-bounded closed free set. We show that this result is sharp. There exists a closed set with bounded sections which has no dominating analytic free set, and there exists a closed set with non-dominating sections which does not have a not σ-bounded analytic free set. Under projective determinacy analytic can be replaced in the above results by projective.
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  9.  7
    Mad Spectra.Saharon Shelah & Otmar Spinas - 2015 - Journal of Symbolic Logic 80 (3):901-916.
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  10.  52
    Meeting Infinitely Many Cells of a Partition Once.Heike Mildenberger & Otmar Spinas - 1998 - Archive for Mathematical Logic 37 (7):495-503.
    We investigate several versions of a cardinal characteristic $ \frak f$ defined by Frankiewicz. Vojtáš showed ${\frak b} \leq{\frak f}$ , and Blass showed ${\frak f} \leq \min({\frak d},{\mbox{\rm unif}}({\bf K}))$ . We show that all the versions coincide and that ${\frak f}$ is greater than or equal to the splitting number. We prove the consistency of $\max({\frak b},{\frak s}) <{\frak f}$ and of ${\frak f} < \min({\frak d},{\mbox{\rm unif}}({\bf K}))$.
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  11.  26
    Additivity of the Two-Dimensional Miller Ideal.Otmar Spinas & Sonja Thiele - 2010 - Archive for Mathematical Logic 49 (6):617-658.
    Let ${{\mathcal J}\,(\mathbb M^2)}$ denote the σ-ideal associated with two-dimensional Miller forcing. We show that it is relatively consistent with ZFC that the additivity of ${{\mathcal J}\,(\mathbb M^2)}$ is bigger than the covering number of the ideal of the meager subsets of ω ω. We also show that Martin’s Axiom implies that the additivity of ${{\mathcal J}\,(\mathbb M^2)}$ is 2 ω .Finally we prove that there are no analytic infinite maximal antichains in any finite product of ${\mathfrak{P}{(\omega)}/{\rm fin}}$.
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  12.  13
    No Tukey Reduction of Lebesgue Null to Silver Null Sets.Otmar Spinas - 2018 - Journal of Mathematical Logic 18 (2):1850011.
    We prove that consistently the Lebesgue null ideal is not Tukey reducible to the Silver null ideal. This contrasts with the situation for the meager ideal which, by a recent result of the author, Spinas [Silver trees and Cohen reals, Israel J. Math. 211 473–480] is Tukey reducible to the Silver ideal.
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  13.  6
    Antichains of perfect and splitting trees.Paul Hein & Otmar Spinas - 2020 - Archive for Mathematical Logic 59 (3-4):367-388.
    We investigate uncountable maximal antichains of perfect trees and of splitting trees. We show that in the case of perfect trees they must have size of at least the dominating number, whereas for splitting trees they are of size at least \\), i.e. the covering coefficient of the meager ideal. Finally, we show that uncountable maximal antichains of superperfect trees are at least of size the bounding number; moreover we show that this is best possible.
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  14.  15
    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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  15.  24
    On the Structure of [Mathematical Formula]-Sets of Reals.Haim Judah & Otmar Spinas - 1995 - Archive for Mathematical Logic 34 (4):301-312.
  16.  6
    Countable Filters on $Omega$.Otmar Spinas - 1999 - Journal of Symbolic Logic 64 (2):469-478.
    Two countable filters on $\omega$ are incompatible if they have no common infinite pseudointersection. Letting $\alpha(P_f)$ denote the minimal size of a maximal uncountable family of pairwise incompatible countable filters on $\omega$, we prove the consistency of t $< \alpha(P_f)$.
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  17.  12
    Canonical Behavior of Borel Functions on Superperfect Rectangles.Otmar Spinas - 2001 - Journal of Mathematical Logic 1 (2):173-220.
    We describe a list of canonical functions from 2 to ℝ such that every Borel measurable function from 2 to ℝ, on some superperfect rectangle, induces the same equivalence relation as some canonical function.
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  18.  5
    On the Structure of $\vec{\delta_4^1}$ -Sets of Reals.Haim Judah & Otmar Spinas - 1995 - Archive for Mathematical Logic 34 (5):301-312.
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  19.  13
    Large Cardinals and Projective Sets.Haim Judah & Otmar Spinas - 1997 - Archive for Mathematical Logic 36 (2):137-155.
    We investigate measure and category in the projective hierarchie in the presence of large cardinals. Assuming a measurable larger than $n$ Woodin cardinals we construct a model where every $\Delta ^1_{n+4}$ -set is measurable, but some $\Delta ^1_{n+4}$ -set does not have Baire property. Moreover, from the same assumption plus a precipitous ideal on $\omega _1$ we show how a model can be forced where every $\Sigma ^1_{n+4}-$ set is measurable and has Baire property.
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  20.  8
    Countable Filters on Ω.Otmar Spinas - 1999 - Journal of Symbolic Logic 64 (2):469-478.
    Two countable filters on ω are incompatible if they have no common infinite pseudointersection. Letting α(P f ) denote the minimal size of a maximal uncountable family of pairwise incompatible countable filters on ω, we prove the consistency of t $.
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