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Let $\kappa$ be an inaccessible cardinal. Let $G$ be generic for $Col(\omega, < \kappa)$, the Levy Collapse.

If $f\in \text{ }^\omega \text{Ord}^{V[G]}$, is $f \in OD_{\text{ }^\omega\omega}^{V[G]}$?

A possibly useful fact is that if $f\in \text{ }^\omega \text{Ord}^{V[G]}$, then there exists a $\lambda < \kappa$ such that $f \in V[G | \lambda]$. However, I do not know how to use this, if relevant at all.

I am interested in this since some people define the Solovay model as $\text{HOD}^{V[G]}_{\text{ }^\omega \text{ORD}}$ and other people use $\text{HOD}_{\text{ }^\omega\omega}^{V[G]}$. I think if the above question has a positive answer, then the two are the same.

Thanks for any help.

$x \in \text{OD}_A$ if $x$ is definable with parameters in $OD \cup A$. $x \in \text{HOD}_A$ if $tc(\{x\}) \subset OD_A$, where $tc$ is the transitive closure.

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  • $\begingroup$ Could you explain what you mean by the subscripts to OD and HOD? $\endgroup$
    – Miha Habič
    Commented Oct 8, 2013 at 2:20
  • $\begingroup$ @MihaHabič I added the definitions. $\endgroup$
    – William
    Commented Oct 8, 2013 at 2:27
  • $\begingroup$ @Miha: Jech uses $\mathsf{[H]OD}(A)$ for that notation. William: And others define the Solovay model as $L(\Bbb R)$, as well. $\endgroup$
    – Asaf Karagila ♦
    Commented Oct 8, 2013 at 6:10

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The answer is negative, and those two models are not necessarily the same. The basic problem is that one might have an $\omega$-sequence of ordinals very high up, above $\kappa$, and if this sequence isn't sufficiently definable in $V$, then it will not be in $\text{HOD}^{V[G]}_{{}^\omega\omega}$, but of course it is in $\text{HOD}^{V[G]}_{{}^\omega\text{Ord}}$.

For example, start with a model where there is a measurable cardinal $\delta$ above $\kappa$ in $V_0$, and let $V=V_0[s]$ be obtained by Prikry forcing at $\delta$ to add a new cofinal $\omega$-sequence $s$ in $\delta$. The forcing to $V[G]$ will be small relative to $\delta$, and we can view $V[G]$ as $V_0[G][s]$, and we can in effect interchange the order of forcing, because the old measure generates a unique new measure, and so $s$ is $V_0[G]$-generic. One can show that after Prikry forcing, the generic sequence is not definable from reals and ordinal parameters, and so $s$ is not in $\text{HOD}^{V[G]}_{{}^\omega\omega}$. But it is an $\omega$-sequence of ordinals, and so it shows the two models are different in this case.

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    $\begingroup$ Probably this kind of question is better placed at MathOverflow. $\endgroup$
    – JDH
    Commented Oct 8, 2013 at 2:52
  • $\begingroup$ Thanks. I was hoping that I could use the proof that $\text{HOD}_{\text{ }^\omega\text{ORD}}^{V[G]} \models DC$ from Jech's book to show $DC$ holds in $\text{HOD}_{\text{ }^\omega\omega}$. However, it seems specific to $\text{ }^\omega \text{ORD}$. Do you know if $DC$ holds in $\text{HOD}_{\text{ }^\omega\omega}^{V[G]}$? $\endgroup$
    – William
    Commented Oct 8, 2013 at 3:35

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