I have a question about the proof of the following:
(Lemma 37) Assume $\mathbf V = \mathbf L$, and let $\kappa$ be a cardinal. Then $\mathcal P (\kappa ) \subseteq L_{\kappa^+}$.
Assume we have proved the following:
(Claim 38) There exists a finite subset $T$ of $ZF + (\mathbf V = \mathbf L)$ such that whenever $x$ is transitive and $\langle x, \overline{\in} \rangle \models T$ then there exists $\gamma \in \mathbf{ON}$ such that $x = L_\gamma$.
(Lemma 35) Let $\alpha \in \mathbf{ON}$ and $x, a_0, \dots , a_{k-1} \in L_\alpha$ and let $\varphi(z, a_0, \dots ,a_{k-1})$ be a formula of $L_S$. Then there exists a $\beta > \alpha$ such that for every $z \in x$: $ \langle L_\beta , \overline{\in} \rangle \models \varphi [z, a_0 , \dots, a_{k-1}] \hspace{0.5cm} \text{ iff } \hspace{0.5cm} \mathbf L \models \varphi [z, a_0 , \dots, a_{k-1}] $
The proof in the book starts as follows: Suppose $\mathbf V = \mathbf L$ holds, let $\kappa$ be as in the assumptions of lemma 37 and suppose $\kappa \in \mathcal P ( \kappa )$. Fix an ordinal $\alpha > \kappa$ such that $y \in L_\alpha$. Since the theory $T$ of claim 38 is finite we can form the conjunction $\varphi$ of all sentences in $T$.
Question: But what is $T$? I know that it's the theory from claim 38 but it's not clear to me how we can use an unknown theory to prove lemma 37. And why do we take the conjunction over $T$?
Then the proof proceeds: By lemma 35,
Question: How does $\gamma < \kappa^+$ follow from $(2)$?
Many thanks for your help.