Tameness, Uniqueness and amalgamation

• Published in: arXiv.org
• Main Author: Jarden, Adi
• Format: Paper
• Language: English
• Published: Ithaca Cornell University Library, arXiv.org 19.09.2015
• Subjects: Amalgamation; Existence theorems; Theorem proving; Uniqueness
• ISSN: 2331-8422

We combine two approaches to the study of classification theory of AECs: 1. that of Shelah: studying non-forking frames without assuming the amalgamation property but assuming the existence of uniqueness triples and 2. that of Grossberg and VanDieren: (studying non-splitting) assuming the amalgamation property and tameness. In [JrSh875], we derive a good non-forking \(\lambda^+\)-frame from a semi-good non-forking \(\lambda\)-frame. But the classes \(K_{\lambda^+}\) and \(\preceq \restriction K_{\lambda^+}\) are replaced: \(K_{\lambda^+}\) is restricted to the saturated models and the partial order \(\preceq \restriction K_{\lambda^+}\) is restricted to the partial order \(\preceq^{NF}_{\lambda^+}\). Here, we avoid the restriction of the partial order \(\preceq \restriction K_{\lambda^+}\), assuming that every saturated model (in \(\lambda^+\) over \(\lambda\)) is an amalgamation base and \((\lambda,\lambda^+)\)-tameness for non-forking types over saturated models, (in addition to the hypotheses of [JrSh875]): We prove that \(M \preceq M^+\) if and only if \(M \preceq^{NF}_{\lambda^+}M^+\), provided that \(M\) and \(M^+\) are saturated models. We present sufficient conditions for three good non-forking \(\lambda^+\)-frames: one relates to all the models of cardinality \(\lambda^+\) and the two others relate to the saturated models only. By an `unproven claim' of Shelah, if we can repeat this procedure \(\omega\) times, namely, `derive' good non-forking \(\lambda^{+n}\) frame for each \(n<\omega\) then the categoricity conjecture holds. Vasey applies one of our main theorems in a proof of the categoricity conjecture under the above `unproven claim' of Shelah and more assumptions. In [Jrprime], we apply the main theorem in a proof of the existence of primeness triples.