Advances in Cardinal Arithmetic

@article{Shelah1993AdvancesIC,
  title={Advances in Cardinal Arithmetic},
  author={Saharon Shelah},
  journal={arXiv: Logic},
  year={1993},
  pages={355-383}
}
  • S. Shelah
  • Published 15 August 2007
  • Mathematics
  • arXiv: Logic
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 we have: {C δ ⋂ α: α ∈ naccC δ } has cardinality < λ. 
A PARTITION RELATION USING STRONGLY COMPACT CARDINALS SH761
If κ is strongly compact and λ > κ and λ is regular (or alternatively cf(λ) ≥ κ), then ( 2 )+ → (λ+ ζ) θ holds for ζ, θ < κ. 2000 Mathematics Subject Classification. 2000 Math Subject Classification:
When Pκ(λ) (vaguely) resembles κ
  • P. Matet
  • Mathematics
    Ann. Pure Appl. Log.
  • 2021
Further cardinal arithmetic
We continue the investigations in the author’s book on cardinal arithmetic, assuming some knowledge of it. We deal with the cofinality of (S≤ℵ0(κ), ⊆) for κ real valued measurable (Section 3),
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
Subsets of cf ( μ ) μ , Boolean Algebras and Maharam measure
The original theme of the paper is the existence proof of “there is η̄ = 〈ηα : α < λ〉 which is a (λ, J)-sequence for Ī = 〈Ii : i < δ〉, a sequence of ideals. This can be thought of as in a
Mathematical Logic
§1 A forcing axiom for λ > א1 fails [The forcing axiom is: if P is a forcing notion preserving stationary subsets of any regular uncountable μ ≤ λ and i is dense open subset of P for i < λ then some
A partition relation using strongly compact cardinals
If κ is strongly compact and A > κ and A is regular (or alternatively cf(A) > κ), then (2 <λ ) + → (λ + ζ) 2 θ holds for ζ, θ < κ.
Reflection Implies the Sch Sh794
We prove that, e.g., if μ > cf(μ) = א0 and μ > 20 and every stationary family of countable subsets of μ reflect in some subset of μ of cardinality א1 then the SCH for μ holds (moreover, for μ, any
Bounds for covering numbers
  • 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.
More on the revised GCH and the black box
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References

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We prove, in an axiomatic way, a compactness theorem for singular cardinals. We apply it to prove that, for singular λ, every λ-free algebra is free; and similar compactness results for transversals
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.
More on cardinal arithmetic
TLDR
This paper deals with variety of problems in pcf theory and infinitary combinatorics, looking at normal filters and prc, measures of the size of [lambda]^{ with |B_i|<mu_i, and the existence of strongly almost disjoint families.
Around classification theory of models
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The core model
The covering lemma for K
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