On the existence of $\kappa$-free abelian groups

Abstract:

It is proved that if ${\aleph _\alpha }$ is a regular cardinal such that there is an ${\aleph _\alpha }$-free abelian group which is not ${\aleph _{\alpha + 1}}$-free, then for every positive integer $n$ there is an ${\aleph _{\alpha + n}}$-free abelian group which is not ${\aleph _{\alpha + n + 1}}$-free. A corollary is that for each positive integer $n$ there is a group of cardinality ${\aleph _n}$ which is ${\aleph _n}$-free but not free. Some results on $\kappa$-free abelian groups which involve notions from logic are also proved.