Pcf theory and cardinal invariants of the reals

The additivity spectrum ADD(I) of an ideal I ⊂ P(I) is the set of all regular cardinals κ such that there is an increasing chain {Aα : α < κ} ⊂ I with ∪α<κAα / ∈ I. We investigate which set A of regular cardinals can be the additivity spectrum of certain ideals. Assume that I = B or I = N , where B denotes the σ-ideal generated by the compact subsets of the Baire space ωω , and N is the ideal of the null sets. We show that if A is a non-empty progressive set of uncountable regular cardinals and pcf(A) = A, then ADD(I) = A in some c.c.c generic extension of the ground model. On the other hand, we also show that if A is a countable subset of ADD(I), then pcf(A) ⊂ ADD(I). For countable sets these results give a full characterization of the additivity spectrum of I: a non-empty countable set A of uncountable regular cardinals can be ADD(I) in some c.c.c generic extension iff A = pcf(A).