Working in $\sf ZF$


Define: $W\_0 = \emptyset \\ W\_{\alpha+1} = H\_{\leq |W\_\alpha|} =\{x \mid \forall y: y \in \operatorname {trcl} (\{x\}) \ |y| \leq |W\_\alpha| \} \\ W\_\lambda= \bigcup W\_{\alpha < \lambda}, \text { for limit ordinal } \lambda$


Where cardinality "| |" is defined after Scott.


This cumulative size hierarchy is also indexed by ordinals.


Now if we define ordinal definable sets in terms of those stages istead of the usual $V\_\alpha$ stages of the cumulative hierarchy. That is, we use exactly the same definition of ordinal definability but only replace the symbol $V$ by $W$. Designate that as $\operatorname {OD}^\*$, then:



> 
> Is $\sf HOD=HOD^\*$?
> 
> 
> 



Is ordinal definability in terms of stages of cumulative size hierarchy equivalent to the usual one?
The answer is yes, in a very general way.


What I claim, first, is that the [Lévy-Montague reflection theorem](https://en.wikipedia.org/wiki/Reflection_principle) holds in ZF for any definable continuous cumulative hierarchical representation of the set-theoretic universe $V$. That is, if you have defined sets $U\_\alpha$ for every ordinal $\alpha$, such that


* the sequence $\alpha\mapsto U\_\alpha$ is definable, without parameters
* the sequence is monotone $\alpha\leq\beta\implies U\_\alpha\subseteq U\_\beta$
* the sequence is continuous, $U\_\lambda=\bigcup\_{\alpha<\lambda}U\_\alpha$ for limit ordinals $\lambda$
* the sequence accumulates to the entire universe, $V=\bigcup\_\alpha U\_\alpha$


Then for every formula $\varphi(x)$ in the first-order language of set theory, there is some ordinal $\alpha$ such that $\varphi$ is absolute between $U\_\alpha$ and $V$. Indeed, there is a closed unbounded class of such ordinals.


The proof is nearly identical to the usual proof of the reflection theorem — these hypotheses are what is used about the $V\_\alpha$ hierarchy.


Once one has reflection, then one can define HOD using the hierarchy, as you suggest, by saying a set $a$ is *ordinal definable*, if there is some $\alpha$ and some ordinal parameters below $\alpha$, such that $a$ is definable in $U\_\alpha$ from those parameters. A set is \*hereditarily ordinal definable, if $a$ and every hereditary member of $a$ is ordinal definable.


In any model $V$ of ZF, if a set is ordinal definable, then by reflection it is ordinal definable in that sense. And conversely, if a set is ordinal definable in that sense, in some $U\_\alpha$ using some formula $\varphi$ with ordinal parameters $\vec \beta$, then in $V$ we may define the set using parameters $\langle\alpha, \vec\beta,\varphi\rangle$ as parameters. (And note the very subtle point that we must sometimes use the formula $\varphi$ itself as a parameter, meaning its Gödel code, since in an $\omega$-nonstandard model the formula $\varphi$ used to define $a$ in $U\_\alpha$ might be nonstandad. I view it as a kind of lucky miracle that ordinal-definability doesn't stumble on this problem.)


So this version aligns with the usual version of HOD.


**Conclusion.** It doesn't matter which definable continuous hierarchy you use when defining HOD. They all give the same class.


Another way to see this is to observe the following:


**Theorem.** For any such hierarchy $U\_\alpha$ as above, there is a closed unbounded class of ordinals $\theta$ for which $$U\_\theta=V\_\theta.$$ In particular, any two such hierarchies agree on a class club.


**Proof.** This is the typical club argument. First, by continuity, the class of such ordinals $\theta$ is closed. And it is unbounded, since we can start in any $U\_{\alpha\_0}$, find a $V\_{\alpha\_1}$ containing all those elements, and then a $U\_{\alpha\_2}$ containing all those elements, and so on, building an alternating chain
$$U\_{\alpha\_0}\subseteq V\_{\alpha\_1}\subseteq U\_{\alpha\_2}\subseteq\cdots$$
If $\theta=\sup\_n\alpha\_n$ is the supremum, then $U\_\theta=V\_\theta$ by continuity. So this is a closed unbounded class. $\Box$


Since reflection also works on a class club, we see in this way that every formula $\varphi$ reflects to a ordinal on which the two hierarchies agree.


