In a constructible universe (ZFC + V=L), is there any known upper bound for the constructibility orders of all elements of the continuum, i.e. some separately described ordinal $\alpha$ such that we can prove $\mathcal{P}(\omega)\subset L_\alpha$ ? For example (under some large cardinal axiom), can it be proven that the first inaccessible cardinal is such an upper bound, or can this cardinal still fail at this ? I intuitively suspect undecidabilities in this matter but am no expert in the field. Thanks.

For simplicity, assume $\mathsf{V=L}$ below.

In fact the situation is as simple as it could possibly be:

For each ordinal $\alpha$, we have $\mathcal{P}(\alpha)\subseteq L_{\vert\alpha\vert^+}$, and moreover for each $\beta<\vert\alpha\vert^+$ there is some $X\subset\alpha$ with $X\in L_{\vert\alpha\vert^+}\setminus L_\beta$.

The second clause follows from the first clause by a simple counting argument (think about the size of $L_\beta$); the first clause is where all the action is. This follows from the **condensation lemma**, which is really the key fact about $L$:

Suppose $M$ is an elementary submodel of $L_\kappa$ for some uncountable cardinal $\kappa$. Then the Mostowski collapse of $M$ is $L_\gamma$ for some $\gamma\le\kappa$.

(Actually this is a *weak* version of the condensation lemma, but it's enough for us.) To see how this can be applied, let's use it to show $\mathbb{R}^L\subseteq L_{\omega_1}$ (which will answer the question in the OP). Fix a real $r\in L$. Taking $\kappa$ large enough, let $M$ be a countable elementary submodel of $L_\kappa$ with $r\in M$. By condensation, let $L_\gamma$ be the Mostowski collapse of $M$. Since $M$ is countable we have $\gamma<\omega_1$. Moreover, the Mostowski collapse map $\mu:M\cong L_\gamma$ doesn't move $r$ (this is a good exercise) so we have $r=\mu(r)\in \mu[M]=L_\gamma$.

More generally, suppose $X\subseteq\alpha$. Fix some uncountable cardinal $\kappa$ with $X\in L_\kappa$, and let $M$ be an elementary submodel of $L_\kappa$ with $\alpha\subseteq M$, $\vert M\vert=\vert L_\alpha\vert$, and $X\in M$. By condensation let $\mu: M\cong L_\gamma$ be the Mostowski collapse map for $M$. By choice of $M$ we have $\mu(X)=X$, so again we get $X=\mu(X)\in \mu[M]=L_\gamma$.

Note that as a corollary this gives $\mathsf{ZFC+V=L\vdash GCH}$.

leastupper bound, so $\mathcal{P}(\omega)\subseteq L_\alpha$ if and only if $\alpha\ge \omega_1$. It follows from a simple cardinal comparison. $\endgroup$ – Hanul Jeon 2 days ago