# Singular in $V$ regular in $HOD$

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

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

(2) $$\kappa$$ is regular in $$HOD$$.

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 $$HOD$$.

• Probably unrelated, but this is certainly true in V=L($\mathbb{R}$)+AD. In particular, $\omega_3$ is singular in V but measurable in HOD. Mar 21, 2016 at 17:46

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 $V$.

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 $HOD$.

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

Homogeneous changes in cofinalities with applications to HOD

• @Rahman.M It doesn't matter. You can always manage all point in C are singular in $V$. May 3, 2016 at 3:49
• What are the large cardinals they're using there? May 3, 2016 at 6:35
• @AsafKaragila I just heard about their result by email correspondence and have not seen their paper. But I think strong type large cardinals are sufficient for it. May 3, 2016 at 7:21