TY - JOUR

T1 - On the arithmetic of density

AU - Kojman, Menachem

N1 - Funding Information:
Research on this paper was partially supported by an Israel Science Foundation grant number 1365/14 .
Publisher Copyright:
© 2016 Elsevier B.V.

PY - 2016/11/1

Y1 - 2016/11/1

N2 - 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.

AB - 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.

KW - Cardinal arithmetic

KW - Density

KW - Generalized Continuum Hypothesis

KW - Silver's theorem

KW - Singular Cardinals Hypothesis

UR - http://www.scopus.com/inward/record.url?scp=84994049565&partnerID=8YFLogxK

U2 - 10.1016/j.topol.2016.08.016

DO - 10.1016/j.topol.2016.08.016

M3 - Article

AN - SCOPUS:84994049565

SN - 0166-8641

VL - 213

SP - 145

EP - 153

JO - Topology and its Applications

JF - Topology and its Applications

ER -