Assuming $V=L$ show that any set of integers $x\in L_{\alpha+1}$ satisfies $x\in L_{\alpha}$ where $\alpha$ is countable.

Question

I can show that every set of integers is constructed at some countable stage of $L$. But apparently it's also true that any set of integers $x\in L_{\alpha+1}$ satisfies $x\in L_{\alpha}$ where $\alpha$ is countable.

I may be missing something obvious, but why is this true?

Answer 1

This is patently false. $\omega\in L_{\omega+1}$ but $\omega\notin L_\omega$. Or, $\varnothing\in L_1$ but $\varnothing\notin L_0$.

Generally, since limit steps do not add new sets, your claim would imply that the constructible hierarchy stabilized $\mathcal P(\omega)$ at the start, which is $\varnothing$. So... that's false.