Special subsets of the reals and tree forcing notions

Abstract

Abstract. We study relationships between classes of special subsets of the reals (e.g. meager-additive sets, γ-sets, C -sets, λ-sets) and the ideals related to the forcing notions of Laver, Mathias, Miller and Silver.

There are no very meager sets in the model in which both the Borel Conjecture and the dual Borel Conjecture are true

“…(5) As always, this gives a contradiction: Let G be R-generic over V and contain y, and let H β be P β -generic over V [G] and contain p; then M y [H y β ] thinks that X y T + (2 ω \ Z ∇ ) (where T isṪ x evaluated by H y β ); so there is an x ∈ X y which is not in T + (2 ω \ Z ∇ ); but x ∈ X y ⊆ X , and ∼ T is forced to be the same asṪ x (cf. (3)), contradicting (6).…”

“…The notion of very meager does not appear often in the literature; the only published reference for the definition we are aware of is [, Definition 2.4]. A quite similar notion was considered by Scheepers (cf.…”

Special subsets of the generalized Cantor space and generalized Baire space

“…The following observation in the case of was made in the proof of [25, Theorem 2.1]. Proposition Assume that κ is a strongly inaccessible cardinal.…”

On the Class of Perfectly Null Sets and Its Transitive Version

“…We say that a set X has v 0 property if for every Silver perfect set P , there exists a Silver perfect set Q ⊆ P \ X (see [10] M. Scheepers (see [22]) proved that if X is a measure zero set with s 0 property, and S is a Sierpiński set, then X + S is also an s 0 -set. Therefore, we easily obtain the following proposition.…”