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There are different criteria for building a model $V$ of $ZFC$ which is far from its $HOD$, for example:

$(A)$ Cardinality criteria: For this in a joint work with James Cummings and Sy Friedman, we produced a model $V$ in which for all infinite cardinals $\alpha$ we have $\alpha^+ > (\alpha^+)^{HOD}.$

$(B)$ Having large cardinals: The idea is to build models which contain some very large cardinals but such that their $HOD$ does not contain them. For this see the nice paper Large cardinals need not be large in HOD by Yong Cheng, Sy Friedman and Hamkins.

$(C)$ The continuum function criteria: To idea is to build models in which $2^\kappa$ is large in $V$ but is small in $HOD$ (for this I recently showed that we can have a model in which $GCH$ fails everywhere but its $HOD$ satisfies the $GCH$).

$(D)$ The cofinality criteria: To build models which contain many singular cardinals which are regular in $HOD$. For this see for example Shoshana Friedman's talk given at the 5th European Set Theory Conference (5ESTC).

I wonder to know what other natural criteria one can consider? In other words what other properties one can consider to measure as a witness for having $HOD$ far from $V$?

References for similar works are appreciated.

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Another criterion may be to make HOD far from $V$ with respect to forcing. For example, in our paper

  • G. Fuchs, J.D. Hamkins, J. Reitz, Set-theoretic geology, Annals of Pure and Applied Logic, vol. 166, iss. 4, pp. 464-501, 2015

we prove that any given model $W$ of ZFC can be made into the Mantle and generic Mantle and HOD and generic HOD of another model $V$, which is a class forcing extension of $W$. The HOD of our model $V$ is not a ground model, so this is a sense in which HOD is far from $V$.

We have other models where the HOD of $V$ is the original model $V_0$, but the mantle is large.

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