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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,

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Question: How does $\gamma < \kappa^+$ follow from $(2)$?

Many thanks for your help.

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1 Answer 1

Claim 38 is a form of what is usually called Gödel's condensation lemma. This is useful since it tells us precisely what the transitive elementary submodels of $\mathbf{L}$ are (if at this point you don't know what an elementary submodel is, just skip this). Precisely what $T$ is isn't particularly important; if you know about Gödel functions, $T$ says something like "$x$ is closed under the Gödel functions". It's not that the theory is unknown, it's just that its precise formulation is annoying and irrelevant. What is important is that it is finite and that it characterizes the levels of $\mathbf{L}$. Finiteness comes into play when taking the conjunction; this allows us to deal with the theory as a single sentence and directly apply Lemma 35. Note that if $T$ were infinite, we might have trouble getting a single $\beta$ that works.

As for your second question, (2) implies that $|L_\gamma |=\kappa$. But you should know, or prove, that $|L_\alpha|=|\alpha|$ for infinite $\alpha$.

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