Erratum to “Categoricity in abstract elementary classes with no maximal models” [Ann. Pure Appl. Logic 141 (2006) 108–147]

• Published in: Annals of pure and applied logic Vol. 164; no. 2; pp. 131 - 133
• Main Author: VanDieren, Monica M.
• Format: Journal Article
• Language: English
• Published: Elsevier B.V 01.02.2013
• ISSN: 0168-0072
• DOI: 10.1016/j.apal.2012.09.003

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 Φμ+(Scf(μ)μ+). 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 μ+.