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Diamond, scales and GCH down to \(\aleph _{\omega ^2}\)

Abstract

Gitik and Rinot (Trans Am Math Soc 364(4):1771–1795, 2012) proved assuming the existence of a supercompact that it is consistent to have a strong limit cardinal \(\kappa \) of countable cofinality such that \(2^\kappa =\kappa ^+\), there is a very good scale at \(\kappa \), and \(\diamond \) fails along some reflecting stationary subset of \(\kappa ^+\cap \text {cof}(\omega )\). In this paper, we force over Gitik and Rinot’s model but with a modification of Gitik–Sharon (Proc Am Math Soc 136(1):311, 2008) diagonal Prikry forcing to get this result for \(\kappa =\aleph _{\omega ^2}\).

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Correspondence to Jin Du.

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Du, J. Diamond, scales and GCH down to \(\aleph _{\omega ^2}\). Arch. Math. Logic 58, 427–442 (2019). https://doi.org/10.1007/s00153-018-0633-4

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Keywords

  • Diamond
  • Scales
  • Stationary
  • Reflection
  • Supercompact

Mathematics Subject Classification

  • 03E05
  • 03E35
  • 03E55