Properties of measure and category in generalized Cohen’s and Silver’s forcing. (English) Zbl 0646.03047

Repický, M.

Acta Univ. Carol., Math. Phys. 28, No. 2, 101-115 (1987).

Let J be an ideal on \(\omega\), and let C(J) be the set of (0,1)-valued functions with domain an element of J, ordered by reverse inclusion (i.e. generalized Cohen forcing, as considered by S. Grigorieff [Ann. Math. Logic 3, 363-394 (1971; Zbl 0328.02041)]. The author proves the following theorem. Let M be a transitive model of ZFC with C(J) in M, and let \(G\subseteq C(J)\) be a generic filter over M. The following are equivalent: (a) for every partition \((x_ n;n<\omega)\) of \(\omega\) with all the \(x_ n\) in J, there is a set x in the filter dual to J such that \(| x\cap x_ n| \leq n\) for all \(n<\omega\); (b) every null set of the Cantor space in M[G] is covered by a null set coded in M; (c) \(^{\omega}2\cap M\) is not a null set in \(^{\omega}2\cap M[G]\). There is a similar characterization of ideals J for which \(^{\omega}\omega \cap M\) is a dominating family in \(^{\omega}\omega \cap M[G]\). The paper concludes with some remarks on generalized Silver forcing.

Reviewer: N.H.Williams

MSC:

03E40	Other aspects of forcing and Boolean-valued models
03E05	Other combinatorial set theory

Keywords:

generalized Cohen forcing; transitive model of ZFC; generic filter; partition; null set of the Cantor space; ideals; dominating family; generalized Silver forcing

Citations:

Zbl 0328.02041