Collapsing successors of singulars

  title={Collapsing successors of singulars},
  author={James Cummings},
Let κ be a singular cardinal in V , and let W ⊇ V be a model such that κ+V = λ + W for some W -cardinal λ with W |= cf(κ) 6= cf(λ). We apply Shelah’s pcf theory to study this situation, and prove the following results. 1) W is not a κ+-c.c generic extension of V . 2) There is no “good scale for κ” in V , so in particular weak forms of square must fail at κ. 3) If V |= cf(κ) = א0 then V |= “κ is strong limit =⇒ 2κ = κ+”, and also ωκ ∩W ) ωκ ∩ V . 4) If κ = אω then λ ≤ (2א0 )V . 

Some consequences of reflection on the approachability ideal

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  • J. Krueger
  • Mathematics
    Journal of Symbolic Logic
  • 2003
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Fallen cardinals

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Exact Upper Bounds and Their Uses in Set Theory



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  • W. Woodin
  • Mathematics
    Proceedings of the National Academy of Sciences of the United States of America
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It is shown that if there exists a supercompact cardinal then every set of reals, which is an element of L(R), is the projection of a weakly homogeneous tree. As a consequence of this theorem and

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Shelah's pcf Theory and Its Applications

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