2373. Saharon Shelah Stable frames and weights
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Submission date: 10 April 2023
Abstract:
Was paper 839 in the author's list until winter 2023 when it was divided into
three.
Part I: We would like to generalize imaginary elements, weight of
ortp(a,M,N), P-weight, P-simple types, etc. from
[She90, Ch. III,V,&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 [She09d] as the (bare bones) right
parallel among a.e.c. to superstable (among elementary classes). Here we
consider (μ, λ, κ)-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 cardinals of small
cofinality). But this arises only when we try to lift. For 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.