# Sh:734

- Shelah, S. (2009).
*Categoricity and solvability of A.E.C., quite highly*. In Classification Theory for Abstract Elementary Classes. arXiv: 0808.3023

Ch. IV of [Sh:h] -
Abstract:

We investigate in ZFC what can be the family of large enough cardinals \mu in which an a.e.c. \mathfrak{K} is categorical or even just solvable. We show that for not few cardinals \lambda < \mu there is a superlimit model in \mathfrak{K}_\lambda. Moreover, our main result is that we can find a good \lambda-frame \mathfrak{s} categorical in \lambda such that \mathfrak{K}_\mathfrak{s} \subseteq \mathfrak{K}_\lambda. We then show how to use 705 to get categoricity in every large enough cardinality if \mathfrak{K} has cases of \mu-amalgamation for enough \mu and 2^\mu < 2^{\mu^{+1}} < \ldots < 2^{\mu^{+n}} \ldots for enough \mu. - Version 2023-02-03 (107p)

Bib entry

@inbook{Sh:734, author = {Shelah, Saharon}, title = {{Categoricity and solvability of A.E.C., quite highly}}, booktitle = {{Classification Theory for Abstract Elementary Classes}}, year = {2009}, note = {\href{https://arxiv.org/abs/0808.3023}{arXiv: 0808.3023} Ch. IV of [Sh:h]}, arxiv_number = {0808.3023}, refers_to_entry = {Ch. IV of [Sh:h]} }