The spectrum of independence, II

We study the set sp(i)={|A|:A⊆[ω]^ω 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 ℵ_ω. Moreover, given an arbitrary set Θ of uncountable cardinals, our techniques allow to obtain a cardinal preserving generic extension in which Θ⊆sp(i), thus showing that sp(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 sp(i)=Θ.