# Upper bound for constructibility orders of elements of the continuum

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.

• Your desired upper bound is $\omega_1$, as follows from the condensation lemma – Wojowu Oct 18 '20 at 12:09
• @Wojowu $\omega_1$ is also the least upper bound, so $\mathcal{P}(\omega)\subseteq L_\alpha$ if and only if $\alpha\ge \omega_1$. It follows from a simple cardinal comparison. – Hanul Jeon Oct 18 '20 at 14:59
• It seems worthwhile to mention that the answer given by @Wojowu is the main point in Gödel's proof that V=L implies the continuum hypothesis. – Andreas Blass Oct 18 '20 at 16:29

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 $$\mathsf{ZFC+V=L\vdash GCH}$$.