Definability in $L(\omega_1)$ I'm trying to solve problem II.6.35 in Kunen's book, which asks to prove that if $V=L$, then the set $B$ of $\beta<\omega_1$ such that $L(\beta)\models ZF-P$ and every element of $L(\beta)$ is definable without parameters, is unbounded in $\omega_1$.
I proved in a previous problem that the set of all the elements of $L(\omega_1)$ that are definable without parameters is some $L(\alpha)$ with $\alpha\in B$. The hint provided belows suggest proving the proposition by contradiction, and claims that if $\sup\{\beta+1|\beta\in B\}<\omega_1$ then it belongs to $L(\alpha)$. I don't see why this is true: is "$\beta\in B$" a formula without parameters?
 A: Here is a sketch of why the notion "$\beta\in B$" is definable in $(L(\omega_1),\in)$ without parameters:
Recall that if $V=L$, then $L(\omega_1)=H(\omega_1)$, so that $(L(\omega_1),\in)\models\mathsf{ZF-P}$. As $\mathsf{HF}\subseteq L(\omega_1)$, we can see in $L(\omega_1)$ the set of all formulas(without parameters) of set theory as certain set $F\subseteq \mathsf{HF}$, for instance using Gödel's numbering. Furthermore there exists a formula $\Phi(x,y,z)$ without parameters such that for all $b\in L(\omega_1)$, $a_1,\ldots,a_n\in b$ and any formula $\varphi(x_1,\ldots,x_n)$ of set theory, the relation $(b,\in\cap b)\models\varphi(a_1,\ldots,a_n)$ holds iff the sentence $\Phi(b,(a_1,\ldots,a_n),\ulcorner\varphi\urcorner)$ is true in $L(\omega_1)$.
As the set of axioms of $\mathsf{ZF-P}$ is recursive, there is a formula $\Psi(x)$ without parameters such that for any $a\in L(\omega_1)$, $L(\omega_1)\models \Psi(a)$ iff $a\in F$ and $a=\ulcorner\varphi\urcorner$ for some $\varphi\in\mathsf{ZF-P}$.
Now consider the following formulas:
$\varphi_1(x):=``x$ is an ordinal$"\land\forall y(y=L(x)\longrightarrow\forall z\in y\exists w(``w$ is a formula of one variable $t$ with no parameters$"\land``(y,\in\cap y)\models w(z)\land\exists !tw(t)"))$.
$\varphi_2(x):=``x$ is an ordinal$"\land\forall y(y=L(x)\longrightarrow \forall w(\Psi(w)\longrightarrow``(y,\in\cap y)\models w"))$.
As the notions $y=L(x)$ and $``w=\ulcorner\phi(t)\urcorner$ for some formula $\phi$ of set theory of one variable $t"$ can be expressed in $L(\omega_1)$ with a formula with no parameters, we get that both $\varphi_1$ and $\varphi_2$ require no parameters.
Finally, notice that $\varphi_1(x)$ says $ ``x$ is an ordinal and all the elements of $L(x)$ can be defined in this model without parameters$"$, and $\varphi_2(x)$ says $``x$ is an ordinal and $L(x)$ is a model of $\mathsf{ZF-P}"$, thus for any ordinal $\beta<\omega_1$ $$\beta\in B\Leftrightarrow L(\omega_1)\models\varphi_1(\beta)\land\varphi_2(\beta).$$ 

Notice that $\gamma:=\sup\{\beta+1:\beta\in B\}$ is in $L(\alpha)$, as $\gamma$ is the least ordinal $\xi$ such that $L(\omega_1)\models\neg(\varphi_1(\beta)\land\varphi_2(\beta))$ for all $\beta\geq\xi$, and thus $\gamma$ is definable in $L(\omega_1)$ without parameters, i.e., $\gamma\in L(\alpha)$.
