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:
Theorem 0.1. Suppose\(2^{\lambda }< 2^{\lambda ^{+}} < 2^{\lambda ^{++}}\)and\(2^{\lambda ^{+}} > \lambda ^{++}\). If \(\mathbb {I}(\mathbf {K}, \lambda ) = \mathbb {I}(\mathbf {K}, \lambda ^{+}) = 1 \le \mathbb {I}(\mathbf {K}, \lambda ^{++}) < 2^{\lambda ^{++}}\)and\(\mathbf {K}\)is\((\lambda , \lambda ^+)\)-tame, then\(\mathbf {K}_{\lambda ^{+++}} \ne \emptyset \).
The proof presented clarifies some of the details of the main theorem of Shelah (Isr J Math 126:29–128, 2001) and avoids using the heavy set-theoretic machinery of Shelah (Classification theory for abstract elementary classes, College Publications, Charleston, 2009 [§VII]) by replacing it with tameness.
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Notes
In Sect. 3.1 we present a more detailed discussion regarding the implications in the other direction.
See [17, §VII.0.4] for a definition of \(\mu _{unif}\) and some of its properties.
Combining further results of Shelah, [21, 7.1] actually gets a good \(\lambda \)-frame and a good \(\lambda ^+\)-frame.
Shelah shows in [17, §VI.7.4] that under additional hypothesis weak density implies density.
It is clear that a \(good^{-(St,Lc)} \lambda \)-frame is stronger than a w-good \(\lambda \)-frame. It is suspected that symmetry does not follow from the other axioms of a good \(\lambda \)-frame, so we suspect that \(good^{-(St,Lc)} \lambda \)-frames are strictly stronger than w-good \(\lambda \)-frames. The reason we do not mention \(good^{-(St,Lc)} \lambda \)-frames until this point is because they are simply a technical tool developed in [11] to encompass both semi-good frames and almost-good frames.
As mentioned in the introduction, Shelah claims the same conclusion from fewer assumptions (see Fact 1.1 and the two paragraphs above it).
In [17, §VI.8.3] Shelah shows, under the hypothesis of Fact 1.1, that \(\mathfrak {s}_{min}\) is an almost good \(\lambda \)-frame. The reason we only show that \(\mathfrak {s}_{min}\) is a w-good \(\lambda \)-frame is because by Sect. 3 this is enough to get a model of size \(\lambda ^{+++}\) and because the known proofs of the other properties use the machinery of [17, §VII] which we avoid.
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Mazari-Armida, M. Non-forking w-good frames. Arch. Math. Logic 59, 31–56 (2020). https://doi.org/10.1007/s00153-019-00677-8
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DOI: https://doi.org/10.1007/s00153-019-00677-8