### Uniformization

by Shelah. [Sh:486]

For a given stationary set S of countable ordinals we
prove (in ZFC) the equivalence of a set theoretic statement and an
abelian group theoretic statement. The first is ``every S-ladder
system has aleph_0-uniformization''. The second says that every
aleph_1-free (abelian) group of cardinality aleph_1 with non
freeness invariant subseteq S is a Whitehead group and is even
co-separable (i.e. Ext (G, Z)^omega =0); if S is not
co-stationary the family of groups is somewhat more restricted. But
only very elementary knowledge of abelian group theory and set
theory is required. This solves problems from Eklof and Mekler's
book.

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