Is “the least $\sigma$ with the property that $L_{\sigma} \prec_1 L$” equal to “the smallest $\sigma$ such that $L_{\sigma} \preceq_1 L$”?

Question

The ordinal $\alpha$ is the least $\Sigma_1$-stable ordinal. It is the least $\sigma$ with the property that $L_{\sigma} \prec_1 L$ or equivalently least with the property that every $\Sigma_1$-statement that is true in $L$ is already true in $L_{\sigma}$.

This ordinal is mentioned in the second paragraph of the Lemma 3 in the paper “The recognizability strength of infinite time Turing machines with ordinal parameters” [Merlin Carl, Philipp Schlicht] and in the Definition 3.1 in the paper “Recognizable sets and Woodin cardinals: Computation beyond the constructible universe” [Merlin Carl, Philipp Schlicht, Philip Welch].

The ordinal $\beta$ is equal to #2.25 in the document “A zoo of ordinals” [David A. Madore]:

The smallest stable ordinal $\sigma$, i.e., the smallest $\sigma$ such that $L_{\sigma} \preceq_1 L$, or equivalently $L_{\sigma} \preceq_1 L_{\omega_1}$. The set $L_{\sigma}$ is the set of all $x$ which are $\Sigma_1$-definable in $L$ without parameters.

This is also the smallest $\Sigma_2^1$-reflecting ordinal.

Are these two ordinals equal to each other? If not, then which ordinal is larger? If yes, then why is $\alpha$ described with the “$\prec_1$” symbol in $L_{\sigma} \prec_1 L$, but $\beta$ is described with the “$\preceq_1$” symbol in $L_{\sigma} \preceq_1 L$?

Answer 1

As Andreas Lietz stated, the only difference between "$X\prec_1 Y$ and "$X\preccurlyeq_1Y$" is that the former requires $X\not=Y$ while the latter allows $X=Y$. So they don't behave differently in this situation (in particular, note that $L_\alpha\not=L$ for every $\alpha$ since $L_\alpha$ is a set but $L$ is a proper class).

The choice of one over the other just reflects a notational preference on the authors' part: do we use "$\prec_1$" since it conveys slightly more information "up front" or do we use "$\preccurlyeq_1$" since that extra information is really trivial anyways and the more flexible notation is generally nicer? (Personally, I lean on the side of "$\preccurlyeq_1$" in all situations where it works, but that's just me.)