  We show that allowing the metric dimension of a space to be independent of
its KO-dimension and turning the finite noncommutative geometry F-- whose
product with classical 4-dimensional space-time gives the standard model
coupled with gravity--into a space of KO-dimension 6 by changing the grading on
the antiparticle sector into its opposite, allows to solve three problems of
the previous noncommutative geometry interpretation of the standard model of
particle physics:
  The finite geometry F is no longer put in "by hand" but a conceptual
understanding of its structure and a classification of its metrics is given.
  The fermion doubling problem in the fermionic part of the action is resolved.
  The spectral action of our joint work with Chamseddine now automatically
generates the full standard model coupled with gravity with neutrino mixing and
see-saw mechanism for neutrino masses. The predictions of the Weinberg angle
and the Higgs scattering parameter at unification scale are the same as in our
joint work but we also find a mass relation (to be imposed at unification
scale).
