TITLE: How would the constructible hierarchy change if you remove parameters?
QUESTION [7 upvotes]: The constructible hierarchy is defined as follows.  $L_0=\varnothing$.  For any ordinal $\beta$, $L_{\beta+1}(X)=Def(L_{\beta})$, where $Def(X)$ is the set of all subsets of $X$ which are first-order definable using elements of $X$ as parameters.  For any limit ordinal $\gamma$, $L_\gamma=\cup_{\beta<\gamma}L_\beta$. Finally, $L=\cup_\alpha L_\alpha$.
My question is, what if instead of taking the set of all subsets definable using parameters, we take the set of all subsets definable without parameters?  How much would the constructible universe shrink?

REPLY [2 votes]: This question shows up as Exercise 23 at the end of Chapter VI in the old version of Kunen's textbook. The claim in the textbook is that the structure defined in the way suggested by the OP is just $L$. I do not know how to show this, but I'm providing a partial answer here to bump this question up.
More specifically, I claim the following: for any $\alpha$ and any $x\in L_{\alpha+1}$, $x$ is definable over $L_\alpha$ with only parameters of the form $L_\xi$, for some $\xi<\alpha$.
We will show this by induction on $\alpha$. Suppose this claim holds below $\alpha$, and suppose
$$
x=\{z\in L_\alpha\mid L_\alpha\vDash \varphi(z, p_1,...,p_n)\}
$$
Then we may fix some $\gamma<\alpha$ with $p_1,...,p_n\in L_{\gamma+1}$. But the claim holds for $L_\gamma$, which means the $p_i$'s can be defined over $L_\gamma$ with only parameters of the form $L_\xi$. For each $p_i$, fix a formula $\psi_i(z)$ (with the appropriate parameters plugged in) that defines $p_i$. And let $\psi_i'(z)$ be the modification of $\psi_i(z)$ by bounding all unbounded quantifiers with $L_\gamma$.
Now it follows that each $p_i$ is definable over $L_\alpha$ in the following way:
$$
p_i=\{z\in L_\alpha\mid L_\alpha\vDash (z\in L_\gamma)\wedge \psi_i'(z)\}
$$
And our set $x$ can be defined as follows:
$$
x=\{z\in L_\alpha\mid L_\alpha\vDash \exists p_1...\exists p_n\varphi(z,p_1,...,p_n)\wedge \\(\forall v)(v\in p_1)\leftrightarrow v\in L_\gamma\wedge \psi_1'(v) \wedge...\wedge\\
(\forall v)(v\in p_n)\leftrightarrow v\in L_\gamma\wedge \psi_n'(v) \}
$$
Basically, the idea is that if your parameters can be defined in this restricted way, then so can you.