MAD families and the rationals
Hrušák, Michael
Commentationes Mathematicae Universitatis Carolinae, Tome 42 (2001), p. 345-352 / Harvested from Czech Digital Mathematics Library
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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.

Publié le : 2001-01-01
Classification:  03E05,  03E17,  03E20 
@article{119248,
     author = {Michael Hru\v s\'ak},
     title = {MAD families and the rationals},
     journal = {Commentationes Mathematicae Universitatis Carolinae},
     volume = {42},
     year = {2001},
     pages = {345-352},
     zbl = {1051.03039},
     mrnumber = {1832152},
     language = {en},
     url = {http://dml.mathdoc.fr/item/119248}
}

Hrušák, Michael. MAD families and the rationals. Commentationes Mathematicae Universitatis Carolinae, Tome 42 (2001) pp. 345-352. http://gdmltest.u-ga.fr/item/119248/

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

Bartoszyński T.; Judah H. Set Theory, On the Structure of the Real Line, A K Peters (1995). (1995) | MR 1350295

Baumgartner J.E.; Laver R. Iterated perfect-set forcing, Annals of Mathematical Logic 17 (1979), 271-288. (1979) | MR 0556894 | Zbl 0427.03043

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. | MR 0776619 | Zbl 0561.54004

Hrušák M. Another $\diamondsuit$-like principle, to appear in Fund. Math. | MR 1815092

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). (1990) | MR 1071305 | Zbl 0718.03037

Kunen K. Set Theory. An Introduction to Independence Proofs, North Holland, Amsterdam, 1980. | MR 0597342 | Zbl 0534.03026

Laflamme C. Zapping small filters, Proc. Amer. Math. Soc. 114 535-544 (1992). (1992) | MR 1068126 | Zbl 0746.04002

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. | MR 0763899 | Zbl 0555.03020

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. | MR 0276079 | Zbl 0226.02047

Shelah S. Proper forcing, Lecture Notes in Mathematics, vol. 940, Springer-Verlag, 1982. | MR 0675955 | Zbl 0819.03042

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