14 found
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  1. On ideals of subsets of the plane and on Cohen reals.Jacek Cichoń & Janusz Pawlikowski - 1986 - Journal of Symbolic Logic 51 (3):560-569.
    Let J be any proper ideal of subsets of the real line R which contains all finite subsets of R. We define an ideal J * ∣B as follows: X ∈ J * ∣B if there exists a Borel set $B \subset R \times R$ such that $X \subset B$ and for any x ∈ R we have $\{y \in R: \langle x,y\rangle \in B\} \in \mathscr{J}$ . We show that there exists a family $\mathscr{A} \subset \mathscr{J}^\ast\mid\mathscr{B}$ of power ω (...)
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  2.  40
    Decomposing Borel functions and structure at finite levels of the Baire hierarchy.Janusz Pawlikowski & Marcin Sabok - 2012 - Annals of Pure and Applied Logic 163 (12):1748-1764.
    We prove that if f is a partial Borel function from one Polish space to another, then either f can be decomposed into countably many partial continuous functions, or else f contains the countable infinite power of a bijection that maps a convergent sequence together with its limit onto a discrete space. This is a generalization of a dichotomy discovered by Solecki for Baire class 1 functions. As an application, we provide a characterization of functions which are countable unions of (...)
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  3.  14
    Why Solovay real produces Cohen real.Janusz Pawlikowski - 1986 - Journal of Symbolic Logic 51 (4):957-968.
    An explanation is given of why, after adding to a model M of ZFC first a Solovay real r and next a Cohen real c, in M[ r][ c] a Cohen real over M[ c] is produced. It is also shown that a Solovay algebra iterated with a Cohen algebra can be embedded into a Cohen algebra iterated with a Solovay algebra.
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  4.  52
    Finite support iteration and strong measure zero sets.Janusz Pawlikowski - 1990 - Journal of Symbolic Logic 55 (2):674-677.
    Any finite support iteration of posets with precalibre ℵ 1 which has the length of cofinality greater than ω 1 yields a model for the dual Borel conjecture in which the real line is covered by ℵ 1 strong measure zero sets.
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  5. Q-pointness, p-pointness and feebleness of ideals.Pierre Matet & Janusz Pawlikowski - 2003 - Journal of Symbolic Logic 68 (1):235-261.
    We study the degree of (weak) Q-pointness, and that of (weak) P-pointness, of ideals on a regular infinite cardinal.
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  6.  14
    Every Sierpiński set is strongly meager.Janusz Pawlikowski - 1996 - Archive for Mathematical Logic 35 (5-6):281-285.
  7. Cohen reals from small forcings.Janusz Pawlikowski - 2001 - Journal of Symbolic Logic 66 (1):318-324.
    We introduce a new cardinal characteristic r*, related to the reaping number r, and show that posets of size $ r* which add reals add unbounded reals; posets of size $ r which add unbounded reals add Cohen reals. We also show that add(M) ≤ min(r, r*). It follows that posets of size < add(M) which add reals add Cohen reals. This improves results of Roslanowski and Shelah [RS] and of Zapletal [Z].
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  8.  32
    Two stars.Janusz Pawlikowski & Marcin Sabok - 2008 - Archive for Mathematical Logic 47 (7-8):673-676.
    The authors investigate an operation * on the subsets of ${\mathcal{P}(\mathbb{R})}$ . It is connected with Borel’s strong measure zero sets as well as strongly meager. The results concern the behaviour of the family of countable sets when * is applied.
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  9.  13
    Adding dominating reals with ωω bounding posets.Janusz Pawlikowski - 1992 - Journal of Symbolic Logic 57 (2):540 - 547.
  10.  8
    Adding Dominating Reals With $omega^omega$ Bounding Posets.Janusz Pawlikowski - 1992 - Journal of Symbolic Logic 57 (2):540-547.
  11.  19
    Extending Baire property by uncountably many sets.Paweł Kawa & Janusz Pawlikowski - 2010 - Journal of Symbolic Logic 75 (3):896-904.
    We show that for an uncountable κ in a suitable Cohen real model for any family {A ν } ν<κ of sets of reals there is a σ-homomorphism h from the σ-algebra generated by Borel sets and the sets A ν into the algebra of Baire subsets of 2 κ modulo meager sets such that for all Borel B, B is meager iff h(B) = 0. The proof is uniform, works also for random reals and the Lebesgue measure, and in (...)
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  12.  4
    Mathias forcing and ultrafilters.Janusz Pawlikowski & Wojciech Stadnicki - 2016 - Archive for Mathematical Logic 55 (7-8):857-865.
    We prove that if the Mathias forcing is followed by a forcing with the Laver Property, then any V\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathsf {V}$$\end{document}-q\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathsf {q}$$\end{document}-point is isomorphic via a ground model bijection to the canonical V\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathsf {V}$$\end{document}-Ramsey ultrafilter added by the Mathias real. This improves a result of Shelah and Spinas.
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  13.  4
    Density zero slaloms.Janusz Pawlikowski - 2000 - Annals of Pure and Applied Logic 103 (1-3):39-53.
    We construct a G δ set G ⊆ ω ω ×2 ω with null vertical sections such that each perfect set P ⊆2 ω meets almost all vertical sections of G in the following sense: we can define from P subsets S of ω of density zero such that whenever the section determined by x ∈ ω ω does not meet P , then x ∈ S for all but finitely many i . This generalizes theorems of Mokobodzki and Brendle (...)
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  14.  3
    The diamond covering property axiom.Janusz Pawlikowski - 2016 - Mathematical Logic Quarterly 62 (4-5):407-411.
    The Covering Property Axiom, which attempts to capture some of the combinatorics of the Sacks model, the model obtained from by countable support iteration of length of the Sacks forcing, seems to miss a Suslin tree. We add a diamond polish to the axiom to remedy this.
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