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- W2218097386 abstract "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." @default.
- W2218097386 created "2016-06-24" @default.
- W2218097386 creator A5059037868 @default.
- W2218097386 date "2014-08-14" @default.
- W2218097386 modified "2023-09-27" @default.
- W2218097386 title "Tameness, Uniqueness and amalgamation" @default.
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