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  • $\begingroup$ Dear Joel, Thanks. It seems that if I replace $\mathbb{M}_{\kappa}$ by $\mathbb{N}_{\kappa}$ everywhere the question will be more interesting because it is consistent to have a measurable cardinal $\kappa$ with exactly $\kappa$ many normal measure which says that the existence of a super measurable cardinal (in the sense of measurability of $|\mathbb{N}_{\kappa}|$) is consistent. $\endgroup$
    – user36136
    Commented Oct 17, 2013 at 14:51