Article
Full entry |
PDF
(0.2 MB)
Feedback
PDF
(0.2 MB)
Feedback
Keywords:
maximal almost disjoint family; Cohen; Miller; Sacks forcing; cardinal invariants of the continuum
maximal almost disjoint family; Cohen; Miller; Sacks forcing; cardinal invariants of the continuum
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.
References:
[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
[BJ] Bartoszyński T., Judah H.: Set Theory, On the Structure of the Real Line. A K Peters (1995). MR 1350295
[BL] Baumgartner J.E., Laver R.: Iterated perfect-set forcing. Annals of Mathematical Logic 17 (1979), 271-288. MR 0556894 | Zbl 0427.03043
[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. MR 0776619 | Zbl 0561.54004
[Hr] Hrušák M.: Another $\diamondsuit$-like principle. to appear in Fund. Math. MR 1815092
[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). MR 1071305 | Zbl 0718.03037
[Ku] Kunen K.: Set Theory. An Introduction to Independence Proofs. North Holland, Amsterdam, 1980. MR 0597342 | Zbl 0534.03026
[La] Laflamme C.: Zapping small filters. Proc. Amer. Math. Soc. 114 535-544 (1992). MR 1068126 | Zbl 0746.04002
[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. MR 0763899 | Zbl 0555.03020
[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. MR 0276079 | Zbl 0226.02047
[Sh] Shelah S.: Proper forcing. Lecture Notes in Mathematics, vol. 940, Springer-Verlag, 1982. MR 0675955 | Zbl 0819.03042
[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
Search
Partner of
