Fix a natural number, $n \geq 1$. Consider the class, M, of all sets hereditarily ordinal-definable using some $\Sigma\_n$ formula. Since there is a universal $\Sigma\_n$ formula, M is definable. Is M necessarily a model of ZF? It seems to me that it is closed under Godel operations and almost universal for the same reasons that HOD is, and therefore a model of ZF. But I feel like I'm missing something, since I've never heard anything about this model. 


If it is a model of ZF, where can I learn more about it? Has anybody done any research about it? How does it relate to HOD?


Note: I require $n \geq 1$ because the formula witnessing that HOD is almost universal is $\Sigma\_1$.



$\Sigma_n$ version of HOD
It follows from the [Reflection Principle](http://en.wikipedia.org/wiki/Reflection_principle) that every ordinal definable set is ordinal definable by a $\Sigma\_2$-formula in the language of set theory. Indeed, if $A = \{x : \phi(x,\bar\alpha)\}$, then $$\exists\gamma(\gamma \in \mathrm{Ord} \land \bar\alpha \in \gamma \land \forall x(x \in A \leftrightarrow x \in V\_\gamma \land V\_\gamma \vDash \phi(x,\bar\alpha))).$$ Therefore, there is some $\gamma \in \mathrm{Ord}$ such that $$x \in A \leftrightarrow \exists U(U = V\_\gamma \land x,\bar\alpha \in U \land U \vDash \phi(x,\bar\alpha)),$$ which is $\Sigma\_2$ since $U = V\_\gamma$ is $\Pi\_1$.


