  The additivity spectrum ADD(I) of an ideal I is the set of all regular
cardinals kappa such that there is an increasing chain {A_alpha:alpha<kappa\}
in the ideal I such that the union of the chain is not in 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.
  For countable sets we 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).

Logic