Non-forking w-good frames

TL;DR

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Contribution

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Abstract

We introduce the notion of a w-good $\lambda$-frame which is a weakening of Shelah's notion of a good $\lambda$-frame. Existence of a w-good $\lambda$-frame implies existence of a model of size $\lambda^{++}$. Tameness and amalgamation imply extension of a w-good $\lambda$-frame to larger models. As an application we show: $\textbf{Theorem}$ Suppose $2^{\lambda}< 2^{\lambda^{+}} < 2^{\lambda^{++}}$ and $2^{\lambda^{+}} > \lambda^{++}$. If $I(K, \lambda) = I(K, \lambda^{+}) = 1 \leq I(K, \lambda^{++}) < 2^{\lambda^{++}}$ and $K$ is $(\lambda, \lambda^+)$-tame, then $K_{\lambda^{+++}} \neq \emptyset$. The proof presented clarifies some of the details of the main theorem of [Sh576] and avoids using the heavy set-theoretic machinery of [Sh: h \S VII] by replacing it with tameness.