  We study the Gray index of phantom maps, which is a numerical invariant of
phantom maps. It is conjectured that the only phantom map with infinite Gray
index between finite-type spaces is the constant map. We disprove this
conjecture by constructing a counter example. We also prove that this
conjecture is valid if the target spaces of phantom maps are restricted to
simply connected finite complexes.
  As an application of the counter example we show that $\SNT^{\infty}(X)$ can
be non-trivial for some space $X$ of finite type.
