In the paper “Categoricity in abstract elementary classes with no maximal models”, we address gaps in Saharon Shelah and Andrés Villavecesʼ (1999) proof in [4] of the uniqueness of limit models of cardinality μ in λ-categorical abstract elementary classes with no maximal models, where λ is some cardinal larger than μ. Both [4] (Shelah and Villaveces, 1999) and [5] (VanDieren, 2006) employ set theoretic assumptions, namely GCH and .
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 Villavecesʼ (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 .