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](http://www.ams.org/journals/proc/2008-136-09/S0002-9939-08-09275-7/S0002-9939-08-09275-7.pdf)", Proceedings of the American Mathematical Society 136, No. 9, 3327-3330 (2008), [MR2407099](https://mathscinet.ams.org/mathscinet-getitem?mr=2407099), [Zbl 1145.03030](https://zbmath.org/1145.03030).



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


