MAD families and the rationals

Michael Hrušák

Commentationes Mathematicae Universitatis Carolinae (2001)

  • Volume: 42, Issue: 2, page 345-352
  • ISSN: 0010-2628

Abstract

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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 𝔟 = 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 𝔡 = ω 1 implies 𝔞 = ω 1 is also discussed.

How to cite

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Hrušák, Michael. "MAD families and the rationals." Commentationes Mathematicae Universitatis Carolinae 42.2 (2001): 345-352. <http://eudml.org/doc/248815>.

@article{Hrušák2001,
abstract = {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 $\mathfrak \{b\} =\mathfrak \{c\}$ 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 $\mathfrak \{d\}=\omega _1$ implies $\mathfrak \{a\}=\omega _1$ is also discussed.},
author = {Hrušák, Michael},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {maximal almost disjoint family; Cohen; Miller; Sacks forcing; cardinal invariants of the continuum; maximal almost disjoint family; Cohen forcing; Miller forcing; Sacks forcing; cardinal invariants of the continuum},
language = {eng},
number = {2},
pages = {345-352},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {MAD families and the rationals},
url = {http://eudml.org/doc/248815},
volume = {42},
year = {2001},
}

TY - JOUR
AU - Hrušák, Michael
TI - MAD families and the rationals
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2001
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 42
IS - 2
SP - 345
EP - 352
AB - 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 $\mathfrak {b} =\mathfrak {c}$ 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 $\mathfrak {d}=\omega _1$ implies $\mathfrak {a}=\omega _1$ is also discussed.
LA - eng
KW - maximal almost disjoint family; Cohen; Miller; Sacks forcing; cardinal invariants of the continuum; maximal almost disjoint family; Cohen forcing; Miller forcing; Sacks forcing; cardinal invariants of the continuum
UR - http://eudml.org/doc/248815
ER -

References

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  8. Laflamme C., Zapping small filters, Proc. Amer. Math. Soc. 114 535-544 (1992). (1992) Zbl0746.04002MR1068126
  9. 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. Zbl0555.03020MR0763899
  10. 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. Zbl0226.02047MR0276079
  11. Shelah S., Proper forcing, Lecture Notes in Mathematics, vol. 940, Springer-Verlag, 1982. Zbl0819.03042MR0675955
  12. 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. MR1234290

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