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Joel David Hamkins
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If you know that $\kappa$ has high Mitchell rank, then it follows that there must be many measurables below with high Mitchell rank, since this is a $\Sigma_2$-property, observable in any model with the same $V_{\kappa+2}$. So it reflects below.

For example, if $j:V\to M$ has critical point $\kappa$ and $M$ has all the measures on $\kappa$, then $M$ will notice that $\kappa$ has high Mitchell rank, and so it is true in $M$ that there are such measurable cardinals below $j(\kappa)$, so there must be many such measurable cardinals below $\kappa$ in $V$.

Joel David Hamkins
  • 236.5k
  • 44
  • 777
  • 1.4k