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Suppose that $X$ is a constructible set, denote by $H^{L_{\alpha}}(X)$ the Skolem hull of $X$ in $L_{\alpha}$. Is $H^{L_{\alpha}}(X)$ constructible?

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up vote 4 down vote accepted

I assume by Skolem hull you mean the one obtained by using the Skolem functions that are definable using the well-ordering of $L_\alpha$ (which, to simplify our life, I'll assume is limit, so the well-ordering of $L$, when interpreted inside $L_\alpha$, is precisely the initial segment of this well-ordering that well-orders $L_\alpha$ itself).

The answer to your question is then yes. $H^{L_\alpha}(X)$ is constructible, as it is definable in, say, $L_{\alpha+\omega}$ from $X$ and $L_\alpha$ (and $\models_{L_\alpha}$ and $\lt_{L_\alpha}$).

The bound $\alpha+\omega$ is lazy, of course.

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