  We show that in any invisible axion model due to the effects of effective
non-renormalizable interactions related to an energy scale near the
Peccei-Quinn, grand unification or even the Planck scale, active neutrinos
necessarily acquire masses in the sub-eV range. Moreover, if sterile neutrinos
are also included and if appropriate cyclic $Z_N$ symmetries are imposed, it is
possible that some of these neutrinos are heavy while others are light.
