An ordinal not $\Sigma_1$ stable in alpha must be in the hull of smaller parameters in $L_\alpha$

Question

This concerns an assertion of Sy Friedman in [1], Lemma 2, which claims that, under certain conditions, if $\beta$ is not 0 and not $\Sigma_1$-stable in $\alpha$, i.e. $L_\beta\prec_1 L_\alpha$, then $\beta$ is in the $\Sigma_1$ Skolem-hull of some $\beta'+1$ in $L_\alpha$, for $\beta'<\beta$. This is used to show that the least $\beta$ such that $y\in L_\beta$ (for some fixed $y$) is 0 or $\Sigma_1$-stable in $\alpha$.

The conditions are that $L_\alpha\vDash V=HC$. This allows that Skolem hulls of arbitrary subsets of $L_\alpha$ are transitive, so the above mentioned $\Sigma_1$ hull of $\beta'+1$ is equal to the hull of $\{\beta'\}$. We also have $\Sigma_1$ Skolem functions, so the Skolem hulls are the ranges of the Skolem functions as we would like.

After looking at the apparently simple assertion for some time I can't see why this should be true, and Asaf, who appears to be trying to get me onto Mathoverflow for some reason, said I should post it!

Reference

[1] Friedman, S.D. "Parameter-free uniformisation", Proceedings of the American Mathematical Society 136, No. 9, 3327-3330 (2008), MR2407099, Zbl 1145.03030.

Answer 1

Since $\beta$ is not $\Sigma_1$-stable in $\alpha$, there is some new $\Sigma_1$ fact $\exists\xi\varphi(\xi,\vec a)$ true in $L_\alpha$ but not in $L_\beta$, for some $\vec a\in L_\beta$. It follows that the witness $\xi$, which we may assume without loss is an ordinal, is at least as large as $\beta$. Let $\beta'$ be an ordinal such that $\vec a$ is $\Sigma_1$ definable from $\beta'$ in $L_\beta$. Now, the $\Sigma_1$-hull of $\beta'$ in $L_\alpha$ includes $\vec a$ and therefore also includes the ordinal $\xi$, and since $\xi$ is countable in $L_\alpha$ in light of V=HC there, the hull will include an enumeration of $\xi$ and therefore will include all ordinals up to $\xi$. In particular, it will include $\beta$ as desired.