323 research outputs found
Categoricity and multidimensional diagrams
We study multidimensional diagrams in independent amalgamation in the
framework of abstract elementary classes (AECs). We use them to prove the
eventual categoricity conjecture for AECs, assuming a large cardinal axiom.
More precisely, we show assuming the existence of a proper class of strongly
compact cardinals that an AEC which has a single model of some high-enough
cardinality will have a single model in any high-enough cardinal. Assuming a
weak version of the generalized continuum hypothesis, we also establish the
eventual categoricity conjecture for AECs with amalgamation.Comment: 63 page
Shelah's Categoricity Conjecture from a successor for Tame Abstract Elementary Classes
Let K be an Abstract Elemenetary Class satisfying the amalgamation and the
joint embedding property, let \mu be the Hanf number of K. Suppose K is tame.
MAIN COROLLARY: (ZFC) If K is categorical in a successor cardinal bigger than
\beth_{(2^\mu)^+} then K is categorical in all cardinals greater than
\beth_{(2^\mu)^+}.
This is an improvment of a Theorem of Makkai and Shelah ([Sh285] who used a
strongly compact cardinal for the same conclusion) and Shelah's downward
categoricity theorem for AECs with amalgamation (from [Sh394]).Comment: 19 page
- …