Definition of the set of definitions. A formula $\varphi[x]$ with one free variable $x$ in the language of sets is a _definition in ZF_ if $ZF \vdash \exists y \forall z(\varphi[z] \longleftrightarrow z = y)$. Is there a definition in ZF of the (meta-)set of Gödel numbers of definitions in ZF? I suspect it isn't but I can't find a contradiction in assuming this existence. One idea would be to find a trick to define the truth of sentences in the language of set theory using the definition of the fact of being a definition, but I don't see how.

Um just let $D = \\{ φ : \text{$φ$ is a $1$-parameter formula over ZF} \land \text{ZF} \vdash \exists! x ( φ[x] )\\}$. Then $D$ is the set you are looking for. You would not be able to get a contradiction from this unless your meta-system is inconsistent.

If ZF is inconsistent, then ZF proves every sentence, and any reasonable meta-system can see that, and so $D$ would be the set of all $1$-parameter formulae. However, if ZF is consistent, it may still be $Σ_1$-unsound and think that itself is inconsistent, in which case $D$ is again the set of all $1$-parameter formulae if you are using ZF as your meta-system.