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Assume that $\mathbb{P}\in HOD$ is non-trivial. It is well-known that if $\mathbb{P}$ satisfies some homogeneity properties, then $HOD^{V[G]} \subseteq V$, where $G$ is $\mathbb{P}$-generic over $V$.

Question. Is there a necessary and sufficient condition on $\mathbb{P}$ implying $HOD^{V[G]} \subseteq V$, whenever $G$ is $\mathbb{P}$-generic over $V$?

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  • $\begingroup$ are there any references for this Q, I've looked and like there's 2 papers, both by the same bloke? $\endgroup$
    – JMP
    Oct 5, 2015 at 7:51
  • $\begingroup$ can i answer 'no' $\endgroup$
    – JMP
    Oct 6, 2015 at 12:20
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    $\begingroup$ @JonMarkPerry Sorry, but I do not understand what you say. $\endgroup$
    – Mohammad Golshani
    Oct 6, 2015 at 12:26
  • $\begingroup$ you could say that to any answer! $\endgroup$
    – JMP
    Oct 6, 2015 at 12:27
  • $\begingroup$ but some questions are independence, so you can not say a yes or no answer to them $\endgroup$
    – Mohammad Golshani
    Oct 6, 2015 at 12:29

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