My questions concern the following quote from “The HOD Dichotomy”, page 8. 


"… notice that $\ cof(\omega)\cap\lambda$ belongs to $HOD$ even though it might mean
something else there. Also, $\{S\subseteq\lambda\mid S\in HOD \text{ and S is stationary}\}$ belongs
to $HOD$ even though there might be sets which are stationary in $HOD$
but not actually stationary. In any case, $HOD$ can recognise when a given
$S\in HOD$ is stationary in $V$...”


My three (probably naive) questions are:


1. Given an ordinal $\lambda$, what could $cof(\omega)\cap\lambda$ mean in $HOD$ that it doesn’t mean in $V$?
2. Perhaps a set $C$ could be club in a given ordinal $\kappa$ in $HOD$, and hence a set $S\in HOD$ could be stationary there, though $S$ is not stationary in $V$. Could there be a set $C$ club in some $\kappa$ which is not actually club in $\kappa$?
3. How can $HOD$ recognize when a particular set $S$ is stationary in $V$ if it isn't stationary in $HOD$?


Is there some kind of absoluteness between $HOD$ and $V$ involving club and stationary sets that simply "falls out of" the notion ordinal or hereditarily ordinal definable that I have missed?



Stationary sets in HOD
The broader point here is that $\text{HOD}$ has all the sets of ordinals that are definable in $V$, and in this way it is able to know some things about what is going on in $V$, even if it cannot see the full reasons for those facts. For example, $\text{HOD}$ has the function giving the cofinality in $V$ of any ordinal, simply because this function (up to any given $\lambda$) is ordinal definable in $V$, even when cofinality is not absolute between $\text{HOD}$ and $V$; and similarly, $\text{HOD}$ has the set of its subsets of $\kappa$ that are stationary in $V$, even when stationarity is not absolute between $\text{HOD}$ and $V$. 


In (1), for example, the point is that $\text{cof}(\omega)\cap\lambda$ is the set of ordinals below $\lambda$ that have cofinality $\omega$ in $V$, and this is a definable set which therefore must be in $\text{HOD}$. But there is no reason to think that these ordinals all have cofinality $\omega$ in $\text{HOD}$, since having cofinality $\omega$ is not necessarily absolute between $V$ and $\text{HOD}$. Indeed, one can easily collapse cardinals to $\omega$, which will make many new ordinals of cofinality $\omega$, but since this forcing is weakly homogeneous, the $\text{HOD}$ of the forcing extension will not think they have cofinality $\omega$. So the set $\text{cof}(\omega)\cap\lambda$ is in $\text{HOD}$, but it isn't necessarily the same as $\text{cof}(\omega)^{\text{HOD}}\cap\lambda$, that is, as the set of ordinals below $\lambda$ that have cofinality $\omega$ in $\text{HOD}$. But the former set will include the latter set, because if an ordinal has cofinality $\omega$ in $\text{HOD}$, then it really has cofinality $\omega$.


For (2), being club is absolute between $\text{HOD}$ and $V$, since for $C\subset\kappa$ to be club means that it is closed and unbounded in $\kappa$, and if you think about it, you will realize that $V$ and $\text{HOD}$ cannot disagree about whether a set is closed or whether it is unbounded in $\kappa$. It follows from this that being stationary is downward absolute, since if a set $S$ really meets every club, then it will also meet every club in $\text{HOD}$, since those clubs really are clubs. But being stationary is not necessarily upward absolute, $\text{HOD}$ may have fewer club sets, and in this way it can be easier for a set to be stationary in $\text{HOD}$ than in $V$. This can definitely happen, if you force over $L$, say to destroy the stationarity of the $L$-least stationary co-stationary subset $S\subset\omega\_1$. This forcing is homogeneous, and so the $\text{HOD}$ of the extension $L[G]$ is just $L$ again, where $S$ is stationary, even though it isn't stationary in $L[G]$. 


For (3), what Woodin means is just that the set $\{S\subset\lambda\mid S\in\text{HOD}\text{ and }S\text{ is stationary in }V\}$ is a set in $\text{HOD}$, and so $\text{HOD}$ has access to this set. So $\text{HOD}$ can know which of its stationary sets are actually stationary in $V$, since the set of those sets is definable in $V$ and hence an element of $\text{HOD}$, and $\text{HOD}$ can tell which are the elements of that particular set. 


