### Stable frames and weights

by Shelah. [Sh:839]

Part I: We would like to generalize imaginary elements,
weight of ortp (a,M,N),{bold P}-weight, {bold P}-simple
types, etc. from [Sh:c, Ch.III,V,section 4] to the context of
good
frames. This requires allowing the vocabulary to have predicates
and
function symbols of infinite arity, but it seemed that we do
not
suffer any real loss.
Part II: Good frames were suggested in [Sh:h] as the (bare
bones) right parallel among a.e.c. to superstable (among
elementary classes). Here we consider (mu, lambda, kappa)-frames
as
candidates for being the right parallel to
the class of |T|^+-saturated models of a stable theory (among
elementary classes). A loss as compared to the superstable case
is
that going up by induction on cardinals is problematic (for cardinalsof small cofinality). But this arises only when we try to lift. But
this context we investigate the dimension.
Part III: In the context of Part II, we consider the main
gap problem for the parallel of somewhat saturated model; showing we
are not worse than in the first order case.

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