### Many forcing axioms for all regular uncountable cardinals

by Shelah. [Sh:832]

Our original aim was, in Abelian group theory to
prove the consistency of: lambda is strong limit
singular and for some properties of abelian groups which are
relatives
of being free, the compactness in singular fails. In fact this
should
work for R-modules, etc. As in earlier cases part of the work
is
analyzing how to move between the set theory and the algebra.
Set theoretically we try to force a universe which satisfies
G.C.H.
and diamond holds for many stationary sets but, for every regular
uncountable lambda, in some sense anything which ``may'' hold
for
some stationary set, does hold for some stationary set.
More specifically we try to get a universe
satisfying GCH such that e.g. for regular kappa < lambda there
are
pairs (S,B),S subseteq S^lambda_kappa stationary, B subseteq{{H}}(lambda), which satisfies some pregiven forcing axiom
related to (S,B), (so (lambda backslash S)-complete, i.e. ``trivial
outside S'') underline {but} no more, i.e. slightly
stronger versions fail (for this S). So set theoretically we
try to
get a universe satisfying G.C.H. but still satisfies ``many'',
even for
a maximal family in some sense, of forcing axioms of the form ``for
some stationary'' while preserving GCH. As completion of the work lag
for long, here we deal only with the set theory.

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