$\sf ZF + Def$ is the theory that extends $\mathcal L(=,\in)\_{\omega\_1,\omega}$ with axioms of $\sf ZF$ (written in $\mathcal L(=,\in)\_{\omega, \omega}$) and the axiom of definability:-


$\textbf{Define: } Dx \iff \bigvee x= \{ y \mid \Phi \}$


where $\Phi$ range over all formulas in $\mathcal L(=,\in)\_{\omega, \omega}$ in which only the symbol "$y$" occurs free, and the symbol "$y$" never occurs bound.


**Axiom of definability:** $\forall x Dx$


This theory has its models being exactly the pointwise-definable models of $\sf ZF$ [[Hamkins](https://mathoverflow.net/a/448434/95347)]


Now, if one wants to confine matters to $\mathcal L(=,\in)\_{\omega,\omega}$, i.e. the usual $\textbf{FOL}(=,\in)$, then one can add this rule to $\sf ZFC$:


$\textbf{Definability: }$ if $\phi\_1,\phi\_2, \phi\_3,...$ are all formulas in which only symbol "$y$" occurs free, and "$y$" never occur bound, and that doesn't use the symbol "$x$", and $\psi$ is a formula in which only symbol "$x$" occurs free, and "$x$" never occur bound; then:


$\underline {[i=1,2,3,...; \\ \forall x \, (x=\{y \mid \phi\_i\} \to \psi)]} \\ \forall x: \psi$


In English: if a parameter free formula holds for all parameter free definable sets, then it holds for all sets.


This was proved by [Hamkins](https://mathoverflow.net/a/448132/95347) to be equivalent over $\sf ZFC$ to the set theoretic axiom $\sf V=HOD$.


The finitary version of definability doesn't manage to confine all of its models to be the pointwise-definable models of $\sf ZFC$. [see [here](https://mathoverflow.net/a/448132/95347)]. But, it can be argued to be the finitary parallel of the infinitary version.


My question here is:



> 
> Are there finitary sentences (i.e. in $\mathcal L(=,\in)\_{\omega,\omega}$) that are theorems of $\sf ZF + Def$ yet not provable in $\sf ZFC + \text { Definability rule}$ (or equivalently in $\sf ZFC+[V=HOD])$? Or is the former a conservative extension of the latter?
> 
> 
> 



> 
> If there are such sentences, are there clear examples?
> 
> 
> 



Is ZF + Def a conservative extension of ZFC+HOD? If not, what are counter-examples?
Any model of ${\sf ZFC}+V=\sf HOD$ has an elementary equivalent pointwise definable model.


If $M$ models $V=\sf HOD$, it has a parameter free definable well ordering, for each formula $φ$ consider the Skolem function that gives the least witness using that well ordering, the class $M\_0$ of definable (without parameters) elements of $M$ will be closed under those Skolem functions, so $M\succ M\_0$, because $M\_0$ is elementary substructure of $M$, the definable elements of $M\_0$ are exactly equal to those of $M$, so by construction $M\_0$ is pointwise definable model that is elementary equivalent to $M$.


In other words, $\sf ZFC$+"being pointwise definable" does not prove anything that ${\sf ZFC}+V=\sf HOD$ does not prove in the language of FOST


