  We investigate the extent of second order characterizable structures by
extending Shelah's Main Gap dichotomy to second order logic. For this end we
consider a countable complete first order theory T. We show that all
sufficiently large models of T have a characterization up to isomorphism in the
extension of second order logic obtained by adding a little bit of infinitary
logic if and only if T is shallow superstable with NDOP and NOTOP. Our result
relies on cardinal arithmetic assumptions. Under weaker assumptions we get
consistency results or alternatively results about second order logic with
Henkin semantics.
