More on the revised GCH and the black box

@article{Shelah2006MoreOT,
  title={More on the revised GCH and the black box},
  author={Saharon Shelah},
  journal={Ann. Pure Appl. Log.},
  year={2006},
  volume={140},
  pages={133-160}
}
  • S. Shelah
  • Published 1 July 2006
  • Mathematics
  • Ann. Pure Appl. Log.
ZF+DC+AX_4
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  • A. Liu
  • Mathematics
    Journal of Symbolic Logic
  • 2006
TLDR
Under various assumptions about the sizes of covering families for cardinals below λ, it is proved upper bounds for the covering number COV (λ. ν+. 2) is closely related to the cofinality of the partial order.
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PCF arithmetic without and with choice
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References

SHOWING 1-10 OF 35 REFERENCES
The generalized continuum hypothesis revisited
We can reformulate the generalized continuum problem as: for regular κ<λ we have λ to the power κ is λ, We argue that the reasonable reformulation of the generalized continuum hypothesis, considering
Reflecting stationary sets and successors of singular cardinals
TLDR
It is shown that supercompactness (and even the failure of PT) implies the existence of non-reflecting stationary sets, and that under suitable assumptions it is consistent that REF and there is a κ which is κ+n-supercompact.
Advances in Cardinal Arithmetic
If cfκ = κ, κ + < cfλ = λ then there is a stationary subset S of {δ < λ : cf(δ) = κ} in I[λ]. Moreover, we can find C = 〈C δ : δ ∈ S〉, C δ club of λ, otp(C δ ) = κ, guessing clubs and for each α < λ
SUCCESSOR OF SINGULARS: COMBINATORICS AND NOT COLLAPSING CARDINALS ≤ κ IN (< κ)-SUPPORT ITERATIONS SH667
On the one hand we deal with (< κ)-supported iterated forcing notions which are (E0, E1)-complete, have in mind problems on Whitehead groups, uniformizations and the general problem. We deal mainly
On squares, outside guessing of clubs and I_{
Suppose that lambda = mu^+. We consider two aspects of the square property on subsets of lambda. First, we have results which show e.g. that for aleph_0 <= kappa =cf (kappa)< mu, the equality
Middle diamond
Abstract.Under certain cardinal arithmetic assumptions, we prove that for every large enough regular λ cardinal, for many regular κ < λ, many stationary subsets of λ concentrating on cofinality κ has
A consistency result on weak reflection
In this paper we study the notion of strong non-reflection, and its contrapositive weak reflection. We say theta strongly non-reflects at lambda iff there is a function F: theta ---> lambda such that
Diamonds, uniformization
  • S. Shelah
  • Mathematics
    Journal of Symbolic Logic
  • 1984
Abstract Assume G.C.H. We prove that for singular λ, □λ implies the diamonds hold for many S ⊆ λ+ (including S ⊆ {δ:δ ∈ λ+, cf δ = cf λ}.) We also have complementary consistency results.
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