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Hechler's theorem for the null ideal

Maxim R. Burke, Masaru Kada
2004 Archive for Mathematical Logic  
This is a variation of Hechler's classical result in the theory of forcing, and the statement of the theorem for the meager ideal has been already proved by Bartoszynski and the author.  ...  is a basis of the null ideal of the real line which is order-isomorphic to Q with respect to set-inclusion.  ...  Acknowledgement The author would like to thank Tomek Bartoszyński, Jörg Brendle, Shizuo Kamo and Teruyuki Yorioka for helpful suggestion and discussion concerning this work.  ... 
doi:10.1007/s00153-004-0224-4 fatcat:75ewvgzoerfw3ftpmjwx3w6kme

Hechler's theorem for the meager ideal [article]

Tomek Bartoszynski, Masaru Kada
2003 arXiv   pre-print
We prove the following theorem: For a partially ordered set Q such that every countable subset has a strict upper bound, there is a forcing notion satisfying ccc such that, in the forcing model, there  ...  This is a variation of Hechler's classical result in the theory of forcing.  ...  Acknowledgement The second author would like to thank Prof. Jörg Brendle and Mr. Teruyuki Yorioka for helpful discussion concerning this work.  ... 
arXiv:math/0209086v2 fatcat:2r2irhkvrbd2dkwaz7q4tkfx4e

Hechler's theorem for the meager ideal

Tomek Bartoszyński, Masaru Kada
2005 Topology and its Applications  
We prove the following theorem: For a partially ordered set Q such that every countable subset has a strict upper bound, there is a forcing notion satisfying ccc such that, in the forcing model, there  ...  This is a variation of Hechler's classical result in the theory of forcing.  ...  Acknowledgement The second author would like to thank Prof. Jörg Brendle and Mr. Teruyuki Yorioka for helpful discussion concerning this work.  ... 
doi:10.1016/j.topol.2003.08.028 fatcat:l2ryg53difcobfk5v5qx34wo7i

Higher Dimensional Cardinal Characteristics for Sets of Functions [article]

Corey Bacal Switzer
2021 arXiv   pre-print
Finally, using variations of Cohen, Hechler, and localization forcing I prove a number of consistency results for possible values of the new cardinals.  ...  These 18 form two separate diagrams consisting of 6 and 12 cardinals respectively, each resembling versions of the Cichoń diagram. These ZFC-inequalities are the first main result of the paper.  ...  Finally the author would like to thank the Austrian Science Fund (FWF) for the generous support through grant number Y1012-N35.  ... 
arXiv:1909.07458v2 fatcat:5aalqi6zzjecxnidgx3c4rbbgi

Simple Cardinal Characteristics of the Continuum

Andreas Blass
1994 arXiv   pre-print
We also discuss some characteristics associated with partition theorems and we present, in an appendix, a simplified proof of Shelah's theorem that the dominating number is less than or equal to the independence  ...  We classify many cardinal characteristics of the continuum according to the complexity, in the sense of descriptive set theory, of their definitions.  ...  Specifically, if we replace the ideal L of measure zero sets by such that the relation y ∈ S x is Π 0 2 (or even Π 0 2 in x for each fixed y) then unif for this ideal is ≥ cov(B).  ... 
arXiv:math/9405202v1 fatcat:yes2yedjkfdelg6rf5josfqoiy

Filter-linkedness and its effect on preservation of cardinal characteristics

Jörg Brendle, Miguel A. Cardona, Diego A. Mejía
2020 Annals of Pure and Applied Logic  
This is applied to prove consistency results about Cichoń's diagram (without using large cardinals) and to prove the consistency of the fact that, for each Yorioka ideal, the four cardinal invariants associated  ...  Although Theorems B, D, E and F are covered by this new result, our methods are different, and in particular Theorem B may be useful as a starting point for further separation results for Cichoń's diagram  ...  Acknowledgements The first author was partially supported by Grants-in-Aid for Scientific Research (C) 15K04977 and 18K03398, Japan Society for the Promotion of Science; the second author was supported  ... 
doi:10.1016/j.apal.2020.102856 fatcat:7q4ugdcsq5ckvjs4lgp27sdmiy

Filter-linkedness and its effect on preservation of cardinal characteristics [article]

Jörg Brendle, Miguel A. Cardona, Diego A. Mejía
2018 arXiv   pre-print
This is applied to prove new consistency results about Cichoń's diagram (without using large cardinals) and to prove the consistency of the fact that, for each Yorioka ideal, the four cardinal invariants  ...  We introduce the property "F-linked" of subsets of posets for a given free filter F on the natural numbers, and define the properties "μ-F-linked" and "θ-F-Knaster" for posets in the natural way.  ...  We include the proof for completeness. Property (a) follows from Lemma 5.8(a). To see property (b), let {ċ(α) : α < θ} be a strongly θ-R-unbounded family added by P.  ... 
arXiv:1809.05004v1 fatcat:tlynuhlelrdufil4iwpbshoyqq

Halfway New Cardinal Characteristics [article]

Jörg Brendle, Lorenz Halbeisen, Lukas Daniel Klausner, Marc Lischka, Saharon Shelah
2018 arXiv   pre-print
Based on the well-known cardinal characteristics s, r and i, we introduce nine related cardinal characteristics by using the notion of asymptotic density to characterise different intersection properties  ...  We prove several bounds and consistency results, e. g. the consistency of s < s_1/2 and s_1/2 < non(N), as well as several results about possible values of i_1/2.  ...  We are grateful to Barnabás Farkas, Martin Goldstern and Salome Schumacher for numerous helpful discussions and suggestions. 1 • the ideal N := {A ⊆ 2 ω | λ(A) = 0} of Lebesgue null sets and • the ideal  ... 
arXiv:1808.02442v2 fatcat:jzovydvxv5h6pid2qjetuz2iqa

Alternative Cichoń Diagrams and Forcing Axioms Compatible with CH [article]

Corey Bacal Switzer
2020 arXiv   pre-print
I prove iteration and preservation theorems for both classes and use these to produce many new models of the subcomplete forcing axiom.  ...  As an application I investigate in detail the Cichoń diagram for degrees of constructibility relative to a fixed inner model of ZFC.  ...  The symbols N , M, K denote the null ideal, the meager ideal and the ideal generated by σ-compact subsets of ω ω respectively.  ... 
arXiv:2008.04900v1 fatcat:enltkdvklnfdlhp3wv4x57iaj4

Coherent systems of finite support iterations [article]

Vera Fischer, Sy D. Friedman, Diego A. Mejía, Diana C. Montoya
2017 arXiv   pre-print
Furthermore, we show that this constellation of 7 values is consistent with the existence of a Δ^1_3 well-order of the reals.  ...  We use them to produce models of new constellations in Cichoń's diagram, in particular, a model where the diagram can be separated into 7 different values.  ...  We should point out though, that if we are to construct a 3D-system for the above constellation, in order to increase s, we have to include in the construction, non-definable, ccc posets which adjoin non-restricted  ... 
arXiv:1609.05433v2 fatcat:rcitiqcs45gplgz5fngpozqwyi

Mazur spaces

Albert Wilansky
1981 International Journal of Mathematics and Mathematical Sciences  
This leads to a new classification of compactT2spacesH, those for which the weak*dual ofC(H)is a Mazur space.  ...  space, and whenXis the weak*dual of a locally convex space.  ...  Namely, in standard Hechler's model h = ℵ 1 whereas b = c (see [3] ). Theorem 7.4. Assume h < b.  ... 
doi:10.1155/s0161171281000021 fatcat:ydowxbs6qra5fijrz2dzqpwyky

Matrix iterations with vertical support restrictions [article]

Diego A. Mejía
2018 arXiv   pre-print
of certain type of unbounded families on definable structures and of certain mad families (like those added by Hechler's poset for adding an a.d. family) regardless of the cofinality of their size.  ...  As applications of this method, we show that a large class of FS iterations can preserve the mad family added by Hechler's poset (regardless of the cofinality of its size), and the consistency of a constellation  ...  IN201711, Dirección Operativa de Investigación, Institución Universitaria Pascual Bravo, and by the Grant-in-Aid for Early Career Scientists 18K13448, Japan Society for the Promotion of Science.  ... 
arXiv:1803.05102v2 fatcat:ufk2lod4a5hq3dzuxx6dx733ce

Template iterations and maximal cofinitary groups

Vera Fischer, Asger Törnquist
2015 Fundamenta Mathematicae  
In [2] , Jörg Brendle used Hechler's forcing notion for adding a maximal almost family along an appropriate template forcing construction to show that a (the minimal size of a maximal almost disjoint family  ...  The main result of the present paper is that ag, the minimal size of maximal cofinitary group, can be of countable cofinality.  ...  Using a combinatorial characterization of add(N ) and cof(N ) (the additivity and cofinality of the null ideal) due to Bartoszyński, we obtain that in the final generic extension both of those cardinal  ... 
doi:10.4064/fm230-3-1 fatcat:znnpgvg3fjd4zawms6cu2d5zxe

Template iterations and maximal cofinitary groups [article]

Vera Fischer, Asger Törnquist
2013 arXiv   pre-print
The main result of the present paper is that a_g, the minimal size of maximal cofinitary group, can be of countable cofinality.  ...  To prove this we define a natural poset for adding a maximal cofinitary group of a given cardinality, which enjoys certain combinatorial properties allowing it to be used within a similar template forcing  ...  Using a combinatorial characterization of add(N ) and cof(N ) (the additivity and cofinality of the null ideal) due to Bartoszyński, we obtain that in the final generic extension both of those cardinal  ... 
arXiv:1310.3245v1 fatcat:afzwzmjrprb63mnsxfp6o4zd4a

Some cardinal invariants of the generalized Baire spaces

Amaya Diana Carolina Montoya
2017 unpublished
We point out the similarities and differences with the classical case and explain the limitations of the classical methods when aiming for such generalizations.  ...  The central theme of the research in this dissertation is the well-known Cardinal invariants of the continuum.  ...  Then N is the ideal of null subsets of 2 ω or ω ω with respect to the corresponding measure.  ... 
doi:10.25365/thesis.47024 fatcat:carzzxpf3fav3mspiljb5cwxju