Non-forking w-good frames

  • Marcos Mazari-ArmidaEmail author


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.


Abstract elementary classes Good frames Tameness 

Mathematics Subject Classification

Primary 03C48 Secondary 03C45 03C55 



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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of Mathematical SciencesCarnegie Mellon UniversityPittsburghUSA

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