$\aleph_n$-free modules with trivial dual

In the first part of this paper we introduce a simplified
version of a new Black Box from Shelah [Sh:883] which can be used to
construct complicated aleph_n-free abelian groups for any natural
number n in N . In the second part we apply this prediction
principle to derive for many commutative rings R the existence of
aleph_n-free R-modules M with trivial dual M^*=0, where
M^*=Hom(M,R) . The minimal size of the aleph_n-free
abelian groups constructed below is beth_n, and this lower bound
is also necessary as can be seen immediately if we apply GCH.

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