Let $\mathfrak{t}$ be the least ordinal such that $L\_{\mathfrak{t}}$ has undefinable ordinals; i.e. there is an $\alpha<\mathfrak{t}$ such that $L\_{\mathfrak{t}}$ cannot define $\alpha$.


This ordinal is quite large, but may have countable bounds under certain conditions. Because this ordinal is at most $\omega\_1$ ($L\_{\omega\_1}$ is an uncountable structure and therefore must have undefinable sets in every uncountable subset, like $\omega\_1$), it is definitely definable because its definition only quantifies over formulae bounded in $L\_{\alpha}$, which are overall bounded in $L\_{\omega\_1}$.




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Here's some data on it:


* It must be a limit ordinal; $L\_{\alpha+1}$ defines $\alpha$ (as the largest ordinal) and thus $L\_\alpha$ allowing for statements like "$\varphi^{L\_\alpha}(\beta)$". Therefore, if $\beta\in\alpha$ is definable in $L\_\alpha$ by the formula $\varphi$, then $\beta$ is definable by $\varphi^{L\_\alpha}$.
* It must be larger than $\omega$ because all finite numbers are clearly definable in $L\_\omega=V\_\omega$
* It must be larger than $\zeta$ (the supremum of eventually writable ordinals) because:
	1. Every turing machine is definably encoded as a natural number in $L\_\omega$
	2. If $\alpha>\omega$, then $L\_\alpha$ can define $L\_\omega$
	3. Klev's $\mathcal{O}^{++}$ is therefore definable in $L\_\alpha$ for $\alpha>\omega$ on any $n$ such that $\mathcal{O}^{++}(n)\in L\_\alpha$
	4. If every ordinal below $\mathfrak{t}$ is eventually writable, then for every $\alpha<\mathfrak{t}$, $\alpha$ can be defined in $L\_{\mathfrak{t}}$ as $\mathcal{O}^{++}(n)$ for some $n$
	5. That's a contradiction, so there is some ordinal below $\mathfrak{t}$ which is not eventually writable, one of which must therefore be $\zeta$
* If there is some countable $\alpha$ such that $\mathcal{P}(L\_\alpha)\cap L\models\text{MK}$, for example an $L$-inaccessible $\alpha$, then $\alpha$ is an upper bound to $\mathfrak{t}$ (that is, $L\_\alpha$ has some undefinable ordinals). This is because:
	1. If $\mathcal{P}(L\_\alpha)\cap L\models\text{MK}$, then $\mathcal{P}^{L}(L\_\alpha)\models\text{MK}$. Therefore $L\models\mathcal{P}(L\_\alpha)\models\text{MK}$.
	2. (Working in $L$ for this bullet) Because $\text{MK}$ proves that $V$ has undefinables in every uncountable subclass, $\mathcal{P}(L\_\alpha)$ also satisfies that $V$ has undefinables in every uncountable subclass.
	3. (Still working in $L$) $V^{\mathcal{P}(L\_\alpha)}=L\_\alpha$, and therefore in every $L\_\alpha$-uncountable subset of $L\_\alpha$, for example $\omega\_1^{L\_\alpha}$, there are $L\_\alpha$-undefinables.
	4. (No longer working in $L$) $L\_\alpha^L=L\_\alpha$, so $\omega\_1^{L\_\alpha}$ has some $L\_\alpha$-undefinables, and therefore $\mathfrak{t}$ is smaller than $\alpha$.


*EDIT: Thanks to user Miha Habič (you should check out some of his brilliant work if you haven't already), we know that this ordinal is for sure countable. Any countable ordinal $\alpha$ which is larger than an $L\_{\omega\_1}$-undefinable ordinal with $L\_\alpha\prec L\_{\omega\_1}$ is clearly an upper bound.*


**So here's the question, just how big is it?**


Is this ordinal admissible? Computably inaccessible? Computably mahlo? Just how big is it? Is it larger than $\Sigma$, the least ordinal which is not accidentally writable?



Is the smallest $L_\alpha$ with undefinable ordinals always countable?
${\mathfrak t}$ is the least $\beta$ such that there is a $\gamma<\beta$ with $L\_\gamma \prec L\_\beta$. That ${\mathfrak t} \leq$ the least such $\beta$ is obvious. On the other hand, if $X \subset L\_{\mathfrak t}$ is $\subseteq$-least with $X \prec L\_{\mathfrak t}$, then $X \not= L\_{\mathfrak t}$; hence if $\sigma \colon L\_\gamma \cong X$, then either $\sigma$ is the identity and thus $\gamma<{\mathfrak t}$ (as desired) or $\sigma$ is not the identity, in which case $L\_{{\rm crit}(\sigma)} \prec L\_{{\sigma}({\rm crit}(\sigma))}$, so that ${\mathfrak t} \leq \sigma({\rm crit}(\sigma)) < {\mathfrak t}$ by the other direction (giving a contradiction). In particular, $L\_{\mathfrak t}$ is a model of ZFC${}^-$ (ZFC w/o the power set axiom), in fact of ZFC${}^-$ plus ``every set is at most countable,'' but ${\mathfrak t}$ is not the least $\beta$ such that $L\_\beta$ is a model of ZFC${}^-$ and it is much bigger than $\Sigma$, the supremum of the accidentally writable ordinals.


