Dominating numbers at singular cardinals

Abstract

We study the generalized dominating number \(\mathfrak {d}_{\mu }\) at a singular cardinal \(\mu \) of cofinality \(\kappa \). We prove two basic lower bounds: in \(\text {ZFC}\), \(\text {cf}\left( {[\mu ]^\kappa ,\subseteq }\right) \le \mathfrak {d}_{\mu }\), and under mild cardinal-arithmetic assumptions, \(2^{<\mu } \le \mathfrak {d}_{\mu }\). We also clarify when \(\mathfrak {d}_{\mu }\) can differ from \(2^\mu \): assuming \(\text {GCH}\) and \(\kappa = \text {cf}\left( {\mu }\right) > \omega \), a finite-support iteration of Cohen forcing of length \(\mu ^{++}\) yields \(\mathfrak {d}_{\mu }< 2^\mu \). On the other hand, for \(\kappa = \text {cf}\left( {\mu }\right) = \omega \), natural \(\mu \)-cc posets force \(\mathfrak {d}_{\mu }= 2^\mu .\)