Bartoszynski, T. and S. Shelah, Closed measure zero sets, Annals of Pure and Applied Logic 58 93–110. We study the relationship between the σ-ideal generated by closed measure zero sets and the ideals of null and meager sets. We show that the additivity of the ideal of closed measure zero sets is not bigger than covering for category. As a consequence we get that the additivity of the ideal of closed measure zero sets is equal to the additivity of the (...) ideal of meager sets. (shrink)

We prove that the cofinality of the smallest covering of R by meager sets is bigger than the additivity of measure.

We conclude the discussion of additivity, Baire number, uniformity, and covering for measure and category by constructing the remaining 5 models. Thus we complete the analysis of Cichon's diagram.

We prove that the following are consistent with ZFC. 1. 2 ω = ℵ ω 1 + K C = ℵ ω 1 + K B = K U = ω 2 (for measure and category simultaneously). 2. 2 ω = ℵ ω 1 = K C (L) + K C (M) = ω 2 . This concludes the discussion about the cofinality of K C.

We answer a question of Just, Miller, Scheepers and Szeptycki whether certain diagonalization properties for sequences of open covers are provably closed under taking finite or countable unions.

A set X⊆ℝ is strongly meager if for every measure zero set H, X+H ≠ℝ. Let [Formula: see text] denote the collection of strongly meager sets. We show that assuming [Formula: see text], [Formula: see text] is not an ideal.

We address ZFC inequalities between some cardinal invariants of the continuum, which turned out to be true in spite of strong expectations given by [11].

We study the cardinal invariants of measure and category after adding one random real. In particular, we show that the number of measure zero subsets of the plane which are necessary to cover graphs of all continuous functions may be large while the covering for measure is small.

We construct a model of ZFC satisfying the Dual Borel Conjecture in which there is a set of size ℵ₁ that does not have measure zero.

We show that it is consistent with ZFC that the intersection of some family of less than ultrafilters have measure zero. This answers a question of D. Fremlin.

Abstract.We will construct several models where there are no strongly meager sets of size 2ℵ0.

In this paper we present two consistency results concerning the existence of large strong measure zero and strongly meager sets. RID=""ID="" Mathematics Subject Classification (2000): 03e35 RID=""ID="" The first author was supported by Alexander von Humboldt Foundation and NSF grant DMS 95-05375. The second author was partially supported by Basic Research Fund, Israel Academy of Sciences, publication 658.

We will construct several models where there are no strongly meager sets of size 2ℵ0.

We show that it is consistent with MA + ¬CH that the Forcing Axiom fails for all forcing notions in the class of ωω–bounding forcing notions with norms of [17].