On the Equivalence between Ch and The Existence of Certain I-Luzin Subsets of ℝ
Enrico Zoli · Georgian Mathematical Journal · 2004
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.