On the arithmetic of density

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Abstract

The κ-density of a cardinal μ≥κ is the least cardinality of a dense collection of κ-subsets of μ and is denoted by D(μ,κ). The Singular Density Hypothesis (SDH) for a singular cardinal μ of cofinality cfμ=κ is the equation D‾(μ,κ)=μ+, where D‾(μ,κ) is the density of all unbounded subsets of μ of ordertype κ. The Generalized Density Hypothesis (GDH) for μ and λ such that λ≤μ is:D(μ,λ)={μ if cfμ≠cfλμ+ if cfμ=cfλ. Density is shown to satisfy Silver's theorem. The most important case is: Theorem 2.6 If κ=cfκ<θ=cfμ<μ and the set of cardinals λ<μ of cofinality κ that satisfy the SDH is stationary in μ then the SDH holds at μ. A more general version is given in Theorem 2.8. A corollary of Theorem 2.6 is: Theorem ⊗ If the Singular Density Hypothesis holds for all sufficiently large singular cardinals of some fixed cofinality κ, then for all cardinals λ with cfλ≥k, for all sufficiently large μ, the GDH holds.

Original languageEnglish
Pages (from-to)145-153
Number of pages9
JournalTopology and its Applications
Volume213
DOIs
StatePublished - 1 Nov 2016

Keywords

  • Cardinal arithmetic
  • Density
  • Generalized Continuum Hypothesis
  • Silver's theorem
  • Singular Cardinals Hypothesis

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