Eventual categoricity spectrum and Frames

by Shelah. [Sh:842]

This work aims at doing parallel of [Sh:705] in a wider and axiomatic way. First, we deal with frames (as in [Sh:600], [Sh:705]), starting for convenience with a fractured a.e.c. K . Our basic frames s ``live'' on such K, so on some cardinals, not just one and possibly not even on an interval of cardinals and types are only formal (and amalgamation is not required). We then add axioms as proved to hold in [Sh:705,section 1-section 11] and proved some things. Second, we deal with [stable] (I, s)-systems and the related property up to beauty parallely to [Sh:705, section 12], including a pure existential version. Third, we prove the main-gap for beautiful frames; this exemplifies that for them we have a full-fledge parallel of the theory of superstable elementary classes. Fourth, we hope to start with an a.e.c. definable in L_{kappa, omega}, kappa measurable and get beauty from solvability in not too small a cardinal. Fifth, we hope to prove that the categoricity and/or solvability spectrum contains or is disjoint to an end segment of the class of cardinals.

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