  We answer one of the main questions in generalized descriptive set theory,
the Friedman-Hyttinen-Kulikov conjecture on the Borel reducibility of the Main
Gap. We show a correlation between Shelah's Main Gap and generalized Borel
reducibility notions of complexity. For any $\kappa$ satisfying
$\kappa=\lambda^+=2^\lambda$ and
$2^{\mathfrak{c}}\leq\lambda=\lambda^{\omega_1}$, we show that if $T$ is a
classifiable theory and $T'$ is a non-classifiable theory, then the isomorphism
of models of $T'$ is strictly above the isomorphism of models of $T$ with
respect to Borel-reducibility.
  We also show that the following can be forced: for any countable first-order
theory in a countable vocabulary, $T$, the isomorphism of models of $T$ is
either analytic co-analytic, or analytically-complete.

Logic