The additivity spectrum $ADD\left(ℐ\right)$ of an ideal $ℐ\subset 𝒫\left(I\right)$ is the set of all regular cardinals $\kappa$ such that there is an increasing chain $\{A_{\alpha }:\alpha <\kappa \}\subset ℐ$ with ${\bigcup }_{\alpha <\kappa }{A}_{\alpha }\notin ℐ$. 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 $\sigma$-ideal generated by the compact subsets of the Baire space ${\omega }^{\omega }$, 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 $pcf\left(A\right)=A$, then $ADD\left(ℐ\right)=A$ in some c.c.c generic extension of the...