Categoricity in abstract elementary classes with no maximal models (vol 141, pg 108, 2006)

In the paper "Categoricity in abstract elementary classes with no maximal models", we address gaps in Saharon Shelah and Andres Villaveces' (1999) proof in 141 of the uniqueness of limit models of cardinality mu in lambda-categorical abstract elementary classes with no maximal models, where lambda is some cardinal larger than mu. Both [4] (Shelah and Villaveces, 1999) and [5] (VanDieren, 2006) employ set theoretic assumptions, namely GCH and Phi(mu)+(S-cf(mu)(mu+)). Recently, Tapani Hyttinen pointed out a problem in an early draft of [3] (Grossberg et al., 2011) to Villaveces. This problem stems from the proof in Shelah and Villaveces' (1999) [4] that reduced towers are continuous. Residues of this problem also infect the proof of Proposition II.7.2 in VanDieren (2006) [5]. We respond to the issues in Shelah and Villaveces (1999) [4] and VanDieren (2006) [5] with alternative proofs under the strengthened assumption that the abstract elementary class is categorical in mu(+). (C) 2012 Elsevier B.V. All rights reserved.