Uncountable $\gamma $-sets under axiom ${\rm CPA}_{\rm cube}^{\rm game}$
Volume 176 / 2003
Fundamenta Mathematicae 176 (2003), 143-155
MSC: Primary 03E35; Secondary 03E17, 26A03.
DOI: 10.4064/fm176-2-3
Abstract
We formulate a Covering Property Axiom CPA$_{\rm cube}^{\rm game}$, which holds in the iterated perfect set model, and show that it implies the existence of uncountable strong $\gamma $-sets in ${\mathbb R}$ (which are strongly meager) as well as uncountable $\gamma $-sets in ${\mathbb R}$ which are not strongly meager. These sets must be of cardinality $\omega _1<{\mathfrak c}$, since every $\gamma $-set is universally null, while CPA$_{\rm cube}^{\rm game}$ implies that every universally null has cardinality less than ${\mathfrak c}=\omega _2$. We also show that CPA$_{\rm cube}^{\rm game}$ implies the existence of a partition of ${\mathbb R}$ into $\omega _1$ null compact sets.
