An ordinal $\alpha$ is "meta-definable" by some formula $\varphi$ without free variables if:  

$$
\begin{cases}
L\_\alpha \models\varphi \\
\forall\beta < \alpha \, L\_\beta \not\models \varphi
\end{cases}
$$
Is there an usual terminology for such "meta-definable" ordinals?



Terminology for ordinals whose constructible level is the least one satisfying some formula
Let me say first that your concept is similar in spirit to the notion of *sententially categorical* cardinal appearing in my joint paper


* J. D. Hamkins and R. Solberg, [Categorical large cardinals and the tension between categoricity and set-theoretic reflection](https://arxiv.org/abs/2009.07164), arxiv:[2009.07164](https://arxiv.org/abs/2009.07164), 2020.


We were interested in investigating the circumstances when Zermelo's quasi-categoricity theorem rises to actual categoricity. Namely, a cardinal $\kappa$ is (first-order) *sententially categorical* if there is some first-order sentence $\sigma$ such that $V\_\kappa\models\text{ZFC}\_2+\sigma$, but this is not true in any other $V\_\beta$. This is equivalent to saying that there is a sentence $\sigma$ for which $V\_\kappa\models\text{ZFC}\_2+\sigma$, but this is not true in any smaller $V\_\beta$. These are equivalent since we can simply replace $\sigma$ with the assertion that $\sigma$ holds, but not earlier.


The difference between your notion and ours is that you are using the $L$ hierarchy and do not insist on $\text{ZFC}\_2$, but the background categoricity idea seems fundamentally similar. In our paper, we consider also theory categoricity, second-order sentential categoricity and second-order theory categoricity.


Meanwhile, the same trick about categoricity works for your notion. That is, I claim an ordinal $\alpha$ is metadefinable on your definition if and only if there is some sentence $\varphi$ true in $L\_\alpha$ and not in any other $L\_\beta$ for $\beta\neq\alpha$. The reason is that if $\varphi$ is true for the first time at $L\_\alpha$, then you can replace $\varphi$ with the assertion "$\varphi$ and this is not true earlier," and this will be true only in $L\_\alpha$.


For this reason, what you have is a notion of sentential cateogoricity—you are identifying ordinals $\alpha$ for which $L\_\alpha$ is characterized among all $L\_\beta$ as the only one satisfying a certain sentence. Therefore, unlesss there is already an established terminology for your ordinals coming from fine-structure theory (see below), I would suggest the terminology: $\alpha$ is *sententially categorical* with respect to the constructible hierarchy.


But second, your context is much lower down. All your ordinals, for example, are countable, since by condensation we can take a countable elementary substructure of $L\_\alpha$, which will collapse below $\omega\_1$, but have the same theory. And since there are only countably many metadefinable ordinals, and this is observable in $L$, they are bounded below $\omega\_1^L$.


Indeed, every meta-definable ordinal is exhibiting a $\Sigma\_1$-property, since $L$ can see that $\alpha$ is metadefinable by observing that there is a sentence $\varphi$ true in $L\_\alpha$ but not earlier. Thus, every metadefinable ordinal is below the first $1$-stable ordinal, the smallest ordinal $\delta$ for which $L\_\delta\prec\_{\Sigma\_1}L$.


Set theorists doing fine structure are often looking at the first stage $L\_\alpha$ where a new $\Sigma\_1$ fact becomes true (allowing parameters), and the supremum of these stages is the first $1$-stable ordinal. But you do not allow parameters in your sentence, and so yours is a very special case. The fine-structure analysis also has notions of $n$-stable and so on. I am not sure if they isolate your notion exactly with terminology, but that is where I would look.


