  The consistent histories formulation of the quantum theory of a closed system
with pure initial state defines an infinite number of incompatible consistent
sets, each of which gives a possible description of the physics. We investigate
the possibility of using the properties of the Schmidt decomposition to define
an algorithm which selects a single, physically natural, consistent set. We
explain the problems which arise, set out some possible algorithms, and explain
their properties with the aid of simple models. Though the discussion is framed
in the language of the consistent histories approach, it is intended to
highlight the difficulty in making any interpretation of quantum theory based
on decoherence into a mathematically precise theory.
