# Article

 Title: MAD families and the rationals (English) Author: Hrušák, Michael Language: English Journal: Commentationes Mathematicae Universitatis Carolinae ISSN: 0010-2628 (print) ISSN: 1213-7243 (online) Volume: 42 Issue: 2 Year: 2001 Pages: 345-352 . Category: math . Summary: Rational numbers are used to classify maximal almost disjoint (MAD) families of subsets of the integers. Combinatorial characterization of indestructibility of MAD families by the likes of Cohen, Miller and Sacks forcings are presented. Using these it is shown that Sacks indestructible MAD family exists in ZFC and that $\frak b =\frak c$ implies that there is a Cohen indestructible MAD family. It follows that a Cohen indestructible MAD family is in fact indestructible by Sacks and Miller forcings. A connection with Roitman's problem of whether $\frak d=\omega_1$ implies $\frak a=\omega_1$ is also discussed. (English) Keyword: maximal almost disjoint family; Cohen Keyword: Miller Keyword: Sacks forcing; cardinal invariants of the continuum MSC: 03E05 MSC: 03E17 MSC: 03E20 idZBL: Zbl 1051.03039 idMR: MR1832152 . Date available: 2009-01-08T19:10:29Z Last updated: 2012-04-30 Stable URL: http://hdl.handle.net/10338.dmlcz/119248 . Reference: [BS] Balcar B., Simon P.: Disjoint refinement.in J.D. Monk and R. Bonnet, editors, Handbook of Boolean Algebras, vol. 2, 1989, pp.333-386. MR 0991597 Reference: [BJ] Bartoszyński T., Judah H.: Set Theory, On the Structure of the Real Line.A K Peters (1995). MR 1350295 Reference: [BL] Baumgartner J.E., Laver R.: Iterated perfect-set forcing.Annals of Mathematical Logic 17 (1979), 271-288. Zbl 0427.03043, MR 0556894 Reference: [vD] van Douwen E.: The integers and topology.in Handbook of Set Theoretic Topology (ed. K. Kunen and J. Vaughan), North-Holland, Amsterdam, 1984, pp.111-167. Zbl 0561.54004, MR 0776619 Reference: [Hr] Hrušák M.: Another $\diamondsuit$-like principle.to appear in Fund. Math. MR 1815092 Reference: [JS] Judah H., Shelah S.: The Kunen-Miller chart (Lebesgue measure, the Baire property, Laver reals and preservation theorems for forcing).J. Symb. Logic 55 909-927 (1990). Zbl 0718.03037, MR 1071305 Reference: [Ku] Kunen K.: Set Theory. An Introduction to Independence Proofs.North Holland, Amsterdam, 1980. Zbl 0534.03026, MR 0597342 Reference: [La] Laflamme C.: Zapping small filters.Proc. Amer. Math. Soc. 114 535-544 (1992). Zbl 0746.04002, MR 1068126 Reference: [Mi] Miller A.: Rational perfect set forcing.in J. Baumgartner, D. A. Martin, and S. Shelah, editors, Axiomatic Set Theory, vol. 31 of Contemporary Mathematics, AMS, 19844, pp.143-159. Zbl 0555.03020, MR 0763899 Reference: [Sa] Sacks G.: Forcing with perfect closed sets.in D. Scott, editor, Axiomatic Set Theory, vol. 1 of Proc. Symp. Pure. Math., AMS, 1971, pp.331-355. Zbl 0226.02047, MR 0276079 Reference: [Sh] Shelah S.: Proper forcing.Lecture Notes in Mathematics, vol. 940, Springer-Verlag, 1982. Zbl 0819.03042, MR 0675955 Reference: [St] Steprāns J.: Combinatorial consequences of adding Cohen reals.in H. Judah, editor, Set theory of the reals, Israel Math. Conf. Proc., vol. 6, 1993, pp583-617. MR 1234290 .

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