Previous |  Up |  Next


Sacks forcing; Marczewski's ideal; cardinal invariants
Under Martin's axiom, collapsing of the continuum by Sacks forcing $\Bbb S$ is characterized by the additivity of Marczewski's ideal (see [4]). We show that the same characterization holds true if $\frak d=\frak c$ proving that under this hypothesis there are no small uncountable maximal antichains in $\Bbb S$. We also construct a partition of $^\omega 2$ into $\frak c$ perfect sets which is a maximal antichain in $\Bbb S$ and show that $s^0$-sets are exactly (subsets of) selectors of maximal antichains of perfect sets.
[1] Balcar B., Vojtáš P.: Refining systems on Boolean algebras. in: Set Theory and Hierarchy Theory, V (Proc. Third Conf., Bierutowice, 1976), Lecture Notes in Math. 619, Springer, Berlin, 1977, pp.45-58; MR 58 #16445. MR 0498304
[2] Balcar B., Simon P.: Disjoint refinement. in: Handbook of Boolean Algebras, Vol. 2 (J.D. Monk and R. Bonnet, Eds.), North-Holland, Amsterdam, 1989, pp.333-388. MR 0991597
[3] Hausdorff F.: Summen von $\aleph_1$ Mengen. Fund. Math. 26 (1936), 241-255; Zbl. 014.05402. DOI 10.4064/fm-26-1-241-255 | Zbl 0014.05402
[4] Judah H., Miller A.W., Shelah S.: Sacks forcing, Laver forcing, and Martin's axiom. Arch. Math. Logic 31 (1992), 3 145-161; MR 93e:03074. DOI 10.1007/BF01269943 | MR 1147737 | Zbl 0755.03026
[5] Kechris A.S.: Classical Descriptive Set Theory. Graduate Texts in Mathematics 156, Springer-Verlag, New York, 1995; MR 96e:03057. MR 1321597 | Zbl 0819.04002
[6] Koppelberg S.: Handbook of Boolean Algebras. Vol. 1 (J.D. Monk and R. Bonnet, Eds.), North-Holland, Amsterdam, 1989; MR 90k:06003. MR 0991565 | Zbl 0671.06001
[7] Marczewski (Szpilrajn) E.: Sur une classe de fonctions de W. Sierpiński et la classe correspondante d'ensembles. Fund. Math. 24 (1935), 17-34; Zbl. 0010.19901. DOI 10.4064/fm-24-1-17-34
[8] Miller A.W.: Covering $2^ømega$ with $ømega_1$ disjoint closed sets. The Kleene Symposium (Proc. Sympos., Univ. Wisconsin, Madison, Wis., 1978), Stud. Logic Foundations Math. 101 (J. Barwise, H.J. Keisler, and K. Kunen, Eds.), North-Holland, Amsterdam, 1980, pp.415-421; MR 82k:03083. DOI 10.1016/S0049-237X(08)71271-0 | MR 0591893
[9] Newelski L.: On partitions of the real line into compact sets. J. Symbolic Logic 52 (1997), 2 353-359; MR 88k:03107. DOI 10.2307/2274384 | MR 0890442
[10] Rosłanowski A., Shelah S.: More forcing notions imply diamond. Arch. Math. Logic 35 (1996), 5-6 299-313; MR 97j:03098. MR 1420260
[11] Simon P.: Sacks forcing collapses $\frak c$ to $\frak b$. Comment. Math. Univ. Carolinae 34 (1993), 4 707-710; MR 94m:03084. MR 1263799
[12] Vaughan J.E.: Small uncountable cardinals and topology. in: Open Problems of Topology (J. van Mill and G.M. Reed, Eds.), North-Holland, Amsterdam, 1990, pp.195-218. MR 1078647
Partner of
EuDML logo