Prikry forcing can be used to produce a model $V$ of $ZFC$ such that fo rsome cardinal $\kappa$ we have:


(1) $\kappa$ is singular in $V$ of cofinality $\omega,$


(2) $\kappa$ is regular (and in fact measurable) in $HOD$.


Now my question is can this happen with $\kappa$ having uncountable cofinality in $V$. So



> 
> **Question** Can we find a model $V$ of $ZFC$ which contains a cardinal $\kappa$ so that
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> (1) $\kappa$ is singular of uncountable cofinality in $V$,
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> (2) $\kappa$ is regular in $HOD$.
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> 


Let me explain why Magidor or Radin forcing do not work in general. Let's start with core model $K$ in which $\kappa$ is large enough, and let $V$ be the generic extension obtained by Magidor or Radin forcing to change the cofinality of $\kappa$ to, say, $\omega\_1.$ Let $C$ be the resulting club. We can assume all elements of $C$ were regular in $K$.


**Claim.** $Lim(C) \in HOD,$ where $Lim(C)$ is the set of limit points of $C$.


**Proof.** We have $Lim(C)=\{\alpha \leq \kappa: \alpha$ is singular, but regular in $K \} \in HOD.$


In particular $\kappa$ is singular in $HOD$.



Singular in $V$ regular in $HOD$
The answer to the question is yes. In fact, it is possible to prove something stronger:



> 
> **Theorem** (Omer Ben-Neria, Spencer Unger) Assuming the existence of suitable large cardinals, there exists a model $V$ of $ZFC$ which contains a club of cardinals all of them are regular in $HOD$ of $V$.
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Their work is under preparation.


**Remark.** Their result is optimal in the sense that we can not hope to build a model in which all infinite cardinals are regular in $HOD$, as for example $\aleph\_\omega$ is always singular in $HOD$.




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**Update**: The paper by Omer Ben-Neria and Spencer Unger is now available:



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>  [Homogeneous changes in cofinalities with applications to HOD](http://www.math.ucla.edu/~sunger/Homogeneous.pdf)
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