The spectrum of independence, II

Abstract

We study the set $\text{sp}(\mathfrak{i})=\{|\mathcal{A}|:\mathcal{A}\subseteq {[\omega ]}^{\omega }\text{ is a maximal independent family}\}$, referred to as the spectrum of independence. We develop a forcing notion, which allows us to adjoin a maximal independent family of arbitrary cardinality, and so in particular of cardinality ${\mathrm{\aleph }}_{\omega }$. Moreover, given an arbitrary set Θ of uncountable cardinals, our techniques allow to obtain a cardinal preserving generic extension in which $\mathrm{\Theta }\subseteq \mathfrak{sp}(\mathfrak{i})$, thus showing that $\mathfrak{sp}(\mathfrak{i})$ can be arbitrarily large. For finite Θ, as well as certain countably infinite Θ, we can obtain a precise equality, i.e. models of $\mathfrak{sp}(\mathfrak{i})=\mathrm{\Theta }$.