Forcing a Basis into ${\aleph }_1$-Free Groups

Abstract

In this paper, we address the question of when a non-free ${\aleph }_1$-free group $H$ can be be free in a transitive cardinality-preserving model extension. Using the $\Gamma$-invariant, denoted $\Gamma (H)$, we present a necessary and sufficient condition resolving this question for ${\aleph }_1$-free groups of cardinality ${\aleph }_1$. Specifically, if $\Gamma (H)=[{\aleph }_1]$, then $H$ will be free in a transitive model extension if and only if ${\aleph }_1$ collapses, while for $\Gamma (H)\ne [{\aleph }_1]$ there exist cardinality-preserving forcings that will add a basis to $H$. In particular, for $\Gamma (H)\ne [{\aleph }_1]$, we provide a poset $({\mathcal{P}}_{\mathrm{pb}},\le )$ of partial bases for adding a basis to $H$ without collapsing ${\aleph }_1$.

12 pages