On the arithmetic of density

TL;DR

This paper investigates the properties of density functions related to cardinals, proving theorems that connect the Singular Density Hypothesis to broader density hypotheses, with implications for set theory and cardinal arithmetic.

Contribution

It establishes conditions under which the Singular Density Hypothesis implies the Generalized Density Hypothesis for large cardinals.

Findings

Density satisfies Silver's theorem.

SDH holds at a cardinal if it holds on a stationary set of smaller cardinals.

SDH for large singulars implies GDH for many cardinals.

Abstract

The $\kappa$-density of a cardinal $\mu\ge\kappa$ is the least cardinality of a dense collection of $\kappa$-subsets of $\mu$ and is denoted by $\mathcal D(\mu,\kappa)$. The Singular Density Hypothesis (SDH) for a singular cardinal $\mu$ of cofinality $cf\mu=\kappa$ is the equation $\mathcal D(\mu,\kappa)=\mu^+$. The Generalized Density Hypothesis (GDH) for $\mu$ and $\lambda$ such that $\lambda\le\mu$ is: $\mathcal D(\mu,\lambda)=\mu$ if $cf\mu\not=cf\lambda$ and $\mathcal D(\mu,\lambda)=\mu^+$ if $cf\mu=cf\lambda$. Density is shown to satisfy Silver's theorem. The most important case is: Theorem 2.6. If $\kappa=cf\kappa<\theta=cf\mu<\mu$ and the set of cardinals $\lambda<\mu$ of cofinality $\kappa$ that satisfy the \textsf{SDH} is stationary in $\mu$ then the SDH holds at $\mu$. A more general version is given in Theorem 2.8 A corollary of Theorem 2.6 is: Theorem 3.2 If the Singular Density Hypothesis holds for all sufficiently large singular cardinals of some fixed cofinality $\kappa$, then for all cardinals $\lambda$ with $cf\lambda \ge \kappa$, for all sufficiently large $\mu$, the GDH holds.