Research Training Network in Model Theory
Publications > Preprint server > Preprint Number 922

Preprint Number 922

Previous Next Preprint server

922. Sebastien Vasey
Shelah's eventual categoricity conjecture in tame AECs with primes

Submission date: 14 September 2015.


Two new cases of Shelah's eventual categoricity conjecture are established:

Let $K$ be an AEC which is tame and has primes over sets of the form $M \cup \{a\}$. If $K$ is categorical in a high-enough cardinal, then $K$ is categorical on a tail of cardinals.
We do not assume amalgamation (however the hypotheses imply that there exists a cardinal $\lambda$ so that $K_{\ge \lambda}$ has amalgamation). The result had previously been established when the stronger locality assumptions of full tameness and shortness are also required.
An application of the theorem is that Shelah's categoricity conjecture holds in the context of homogeneous model theory:

Let $D$ be a homogeneous diagram in a first-order theory $T$. If $D$ is categorical in a $\lambda > |T|$, then $D$ is categorical in all $\lambda' \ge \min (\lambda, \beth_{(2^{|T|})^+})$.

Mathematics Subject Classification: 03C48 (Primary), 03C45, 03C52, 03C55 (Secondary)

Keywords and phrases:

Full text arXiv 1509.04102: pdf, ps.

Last updated: September 25 2015 13:06 Please send your corrections to: