2015/07/08 by Grossberg, Rami, VanDieren, Monica, Villaveces, Andres
#FOS: Mathematics #Logic (math.LO)
paper · doi:10.48550/arxiv.1507.02118
We prove: Main Theorem: Let K be an abstract elementary class satisfying the joint embedding and the amalgamation properties with no maximal models of cardinality μ. Let μ be a cardinal above the the Löwenheim-Skolem number of the class. If K is μ-Galois-stable, has no μ-Vaughtian Pairs, does not have long splitting chains, and satisfies locality of splitting, then any two (μ,σ_ℓ)-limits over M, for ℓ∈\1,2\, are isomorphic over M. This theorem extends results of Shelah from \citeSh394, \citeSh576, \citeSh600, Kolman and Shelah in \citeKoSh and Shelah and Villaveces from \citeShVi. A preliminary version of our uniqueness theorem, which was circulated in 2006, was used by Grossberg and VanDieren to prove a case of Shelah's categoricity conjecture for tame abstract elementary classes in \citeGrVa2. Preprints of this paper have also influenced the Ph.D. theses of Drueck \citeDr and Zambrano \citeZa. This paper also serves the expository role of presenting together the arguments in \citeVa1 and \citeVa2 in a more natural context in which the amalgamation property holds and this work provides an approach to the uniqueness of limit models that does not rely on Ehrenfeucht-Mostowski constructions.