On the Equivalence between Ch and The Existence of Certain I-Luzin Subsets of ℝ

Abstract

We extend Rothberger's theorem (on the equivalence between CH and the existence of Luzin and Sierpiński-sets having power 𝔠) and certain paradoxical constructions due to Erdös. More precisely, by employing a suitable σ-ideal associated to the (α, β)-games introduced by Schmidt, we prove that the Continuum Hypothesis holds if and only if there exist subgroups of (ℝ, +) having power 𝔠 and intersecting every “absolutely losing” (respectively, every meager and null) set in at most countably many points.

Set-Theoretic Properties of Schmidt's Ideal

“…Let us now summarize the main results of our paper. After collecting in Theorem 2.2 a few necessary Schmidt's results about S, in the third section we prove that S is G δ -generated (this sharpens Theorem 12 in [23]). We also prove in Theorem 3.3 that S is invariant under diffeomorphisms (our ad hoc proof is simpler than that of Theorem 1 in [21]).…”

“…Proof. It follows from Theorem 2 in [21] (see also [23], Theorem 10) that S is a σ-ideal. Obviously S is proper, uniform and translation invariant.…”

“…By S we denote the family of all losing subsets of R. We call S Schmidt's σideal. Our definition makes sense, for it is shown in [21] (for all real parameters) and in [23] (for the rational ones) that the family of losing sets is indeed a σ-ideal on R.…”

“…In Schmidt's original paper all real parameters α ∈ (0, 1 2 ) and β ∈ (0, 1) are considered. Following [23], here we focus exclusively on rational ones. Observe that in this way we define an ideal possibly larger than the original one (unfortunately we do not know whether this inclusion is strict).…”

“…In our paper (following [23]) we study an ideal S being a minor -but seemingly necessary for obtaining good descriptive results -a modification of the original Schmidt's σ-ideal (for this reason also S will be referred to as Schmidt's ideal). It turns out that S plays in R the same role as M does in 2 N .…”