= Constructible sets of low rank can appear at later stages
{title2=$L\cap V_{\omega+1}\ne L_{\omega+1}$}
The equality $L\cap V_\alpha=L_\alpha$ can fail even at $\alpha=\omega+1$. Inside the <constructible universe>, $L_\omega=V_\omega$ is countable, so its <definable power set> $L_{\omega+1}$ is countable. The <constructible power set> of $\omega$ is uncountable there by the <Cantor theorem>. A real outside $L_{\omega+1}$ remains a constructible subset of $\omega$, hence belongs to $V_{\omega+1}$. The cardinal comparison must be made inside $L$, since its reals can be externally countable.
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