  The main result of the present note states that it is consistent with the ZFC
axioms of set theory (relying on Martin's Maximum MM axiom), that every Asplund
space of density character $\omega_1$ has a renorming with the Mazur
intersection property. Combined with the previous result of Jim\' enez and
Moreno (based upon the work of Kunen under the continuum hypothesis) we obtain
that the MIP normability of Asplund spaces of density $\omega_1$ is undecidable
in ZFC.
