First time I encounter the Constructible universe $L$ and the definition given in Jech is the following:
- $L_0=\emptyset$,
- $L_{\alpha+1}=\operatorname{def}(L_\alpha)$,
- $L_\lambda=\bigcup_{\alpha<\lambda}L_\alpha$.
Where we have $\operatorname{def}(M)=\{X\subseteq M:\text{ X is definable in }(M,\in)\}$
My doubt is what's the cardinality of $L_{\omega+1}$. From the definitions it would seem it's countable since $L_{\omega}$ is countable, so is the number of $n$-tuples and so will all the finite sequences of elements of $L_{\omega}$. We also have the number of $\{\in\}$-formulas is countable so it would seem that $L_{\omega+1}$ must also be countable. Is this correct? If so since $L$ is a model of $ZFC$ which can prove $\mathcal{P}(\omega)\subseteq V_{\omega+1}$ it would seem the constructible universe thinks $L_{\omega+1}$ is uncountable. Is this also correct?
You are correct that $L_{\omega+1}$ is countable, but it's not correct that $L$ thinks it is uncountable. Although $L$ satisfies $V=L$, it does not satisfy $V_\alpha=L_\alpha$ for each $\alpha$. Every set in the $V$ hierarchy eventually shows up in the $L$ hierarchy, but typically at a later stage. So $L_{\omega+1}$ does not contain all of the subsets of $\omega$ that are in $L$; in fact, you have to go all the way up to $L_{\omega_1^L}$ to capture all of the subsets of $\omega$ in $L$.