A Banach space contains either a minimal subspace or a continuum of
incomparable subspaces. General structure results for analytic equivalence
relations are applied in the context of Banach spaces to show that if E0β
does not reduce to isomorphism of the subspaces of a space, in particular, if
the subspaces of the space admit a classification up to isomorphism by real
numbers, then any subspace with an unconditional basis is isomorphic to its
square and hyperplanes and has an isomorphically homogeneous subsequence