TITLE: Ways to define "definability"
QUESTION [12 upvotes]: The notion of a definable set is not expressible in the language of set theory: there is no formula $\delta(x)$ that is equivalent with there being a formula $\phi(y)$ such that $x = \lbrace y : \phi(y)\rbrace$, i.e. for which
$$\delta(x) \leftrightarrow (\exists \phi \in \mathsf{Form})\ x = \lbrace y : \phi(y)\rbrace$$
is provable. (Please forgive my blunt use of the set-builder notation $\lbrace y : \phi(y)\rbrace$. I know there's the rub.)
The same holds for (unrestricted) definitions with parameters: there is no formula $\delta(x)$ that is equivalent with there being a formula $\phi(y,z_1,\dots,z_n)$ such that $x = \lbrace y : \phi(y,z_1,\dots,z_n)\rbrace$ for some $z_1,\dots,z_n$, i.e. for which
$$\delta(x) \leftrightarrow (\exists \phi \in \mathsf{Form})(\exists z_1,\dots,z_n)\ x = \lbrace y : \phi(y,z_1,\dots,z_n)\rbrace$$
is provable.
But the notion of ordinal-definable sets is expressible in the language of set theory (see Jech, p. 194), i.e. there is a formula $\delta(x)$ such that - bluntly said - 
$$\delta(x) \leftrightarrow (\exists \phi \in \mathsf{Form})(\exists z_1,\dots,z_n)\ x = \lbrace y : \phi(y,z_1,\dots,z_n)\rbrace \wedge \mathsf{ON}(z_i)  $$
is provable.
My questions are:


Are there other restrictions on the parameters like $\mathsf{ON}(z_i)$ (the formula that states that $z_i$ is an ordinal)
  that give rise to a definable notion of definability, i.e. a class of
  accordingly definable sets?
Can these restrictions been characterized? What do they have in common, eventually? What is essential?
Are there restrictions $\Psi(\phi)$ on the formulas that comparably give rise to a definable notion of definability, i.e. there would be a formula
  $\delta(x)$ such that 
  $$\delta(x) \leftrightarrow (\exists \phi \in \mathsf{Form})(\exists z_1,\dots,z_n)\ x = \lbrace y : \phi(y,z_1,\dots,z_n)\rbrace \wedge \Psi(\phi)  $$
  is provable. 
Or maybe some mixture $\Psi(\phi,z_1,\dots,z_n)$?


Question 3 includes: Can we - in a Goedelian way - define for a formula (as a set), that some of its arguments must fulfill some other formula?

REPLY [6 votes]: With respect to question 3: one way to do this is to restrict the quantifier complexity of the formulas. That is, for every constant $k$, one can define a formula $\delta_k(x)$ expressing “$x$ is definable by a $\Sigma_k$-formula with parameters”. Here, a formula $\phi(\vec x)$ is $\Sigma_k$ (in the Lévy hierarchy) if it can be written in the form $\exists y_1\,\forall y_2\,\exists y_3\,\dots Qy_k\,\theta(\vec x,\vec y)$, where all quantifiers in $\theta$ are bounded.
$\def\Tr{\mathrm{Tr}}$EDIT: Let me clarify how exactly $\Sigma_k$-formulas can be used to answer question 3. What I wrote above is true, but useless by itself, because definability of sets by formulas allowing arbitrary parameters is not an interesting notion (every set is trivially definable from itself as a parameter). But the Lévy hierarchy provides more than that:

For every $k>0$, there is a truth definition for $\Sigma_k$-formulas, which is itself $\Sigma_k$. That is, there is a $\Sigma_k$-formula $\Tr_k(n,x)$ such that ZF proves
$$\phi(x)\leftrightarrow\Tr_k(\ulcorner\phi\urcorner,x)$$
for every $\Sigma_k$-formula $\phi(x)$, where $\ulcorner\phi\urcorner$ is the Gödel number of $\phi$. This is essentially optimal: if a class of formulas has a truth definition, it is up to equivalence included in some $\Sigma_k$ (because the truth definition itself is in $\Sigma_k$ for some $k$).
If $k>0$, and $X$ is a class of possible parameters which includes $\omega$ and is closed under pairing (that is, $X\times X\subseteq X$), there is a formula $\delta(x)$ which expresses “$x$ is $\Sigma_k$-definable using parameters from $X$”, namely
$$\delta(x)=\exists n\in\omega\,\exists z\in X\,\forall y\,(y\in x\eq\Tr_k(n,\langle y,z\rangle)).$$
(Here, the pairing operation on $X$ does not have to be the standard Kuratowski pair. For example, one can define an injection $\mathrm{Ord}\times\mathrm{Ord}\to\mathrm{Ord}$, which allows us to take $X=\mathrm{Ord}$, so this generalizes ordinal definability. Also, I believe the assumption $\omega\subseteq X$ is redundant as long as $|X|\ge2$, because one can reconstruct a suitable copy of $\omega$ inside $X$ using the pairing operation.)