  Uniform metastable convergence is a weak form of uniform convergence for a
family of sequences. In this paper we explore the way that metastable
convergence stratifies into a family of notions indexed by countable ordinals.
  We give two versions of this stratified family, loosely speaking, they
correspond to the model theoretic and proof theoretic perspectives. For the
model theoretic version, which we call abstract alpha-uniform convergence, we
show that uniform metastable convergence is equivalent to abstract
$\alpha$-uniform convergence for some alpha, and that abstract omega-uniform
convergence is equivalent to uniformly bounded oscillation of the family of
sequences.
  The proof theoretic version, which we call concrete alpha-uniform
convergence, is less canonical (it depends on a choice of ordinal notation),
but appears naturally when "proof mining" convergence proofs to obtain
quantitative bounds.
  We show that these hierarchies are strict by exhibiting a family of which is
concretely alpha+1-uniformly convergent but not abstractly alpha-uniformly
convergent for each alpha<omega_1.
