Under Measurables, HOD_{Ord^{\omega}} is locally arbitrary
For the inner model $\HOD_{\Ord^{\omega}}$, we derive an analogue for Roguski's classical result that without nontrivial assumptions on $V$, $\HOD$ is an arbitrary model of $\ZFC$. For any sentence $\sigma$, letting $\CM$ denote ``class of measurables,'' the following theories are equiconsistent: \begin{enumerate} \item $\ZFC+\CM+ [\HOD_{\Ord^{\omega}}\models (\exists\kappa\,V_{\kappa}\models\sigma)]$; \item $\ZFC+\GCH+\CM+ [\HOD_{\Ord^{\omega}}\models (\SVC\wedge\CM\wedge\exists\kappa\,V_{\kappa}\models\sigma)]$; \item $\ZF+\DC+\CM+ \exists\kappa\,V_{\kappa}\models\sigma$. \end{enumerate}…
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