Suppose $V=L$ + reasonable hypotheses (e.g. "ZFC has a countable transitive model"). Call a countable ordinal $\alpha$ *memorable* if for some countable $\beta$, $\alpha$ is definable *without parameters* in every $L\_\gamma$ with $\beta<\gamma<\omega\_1$.


My question is:



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> Are there uncountably many memorable ordinals?
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Some comments, right off the bat:


* The answer is trivially "no" for *uniformly* memorable ordinals, that is, ordinals which are coboundedly definable without parameters *by the same formula*. However, since the defining formula is allowed to vary with $\gamma$, this doesn't work.
* Towards a positive answer, note that there are uncountably many countable $\theta$ such that $L\_\theta$ is pointwise definable (call such an ordinal "insightful"); this was proved by [Hamkins, Linetsky, and Reitz](https://arxiv.org/pdf/1105.4597v2.pdf). However, this doesn't actually resolve the issue: for a fixed $\alpha$, there may be many countable $\eta>\alpha$ with no insightful ordinals $>\alpha$ which are definable without parameters in $L\_\eta$ (e.g. if there is no greatest insightful ordinal $<\eta$, this seems a distinct possibility).




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Interestingly, this question has possibly interesting variants even if $L$ is a very tiny subclass of $V$! Given any hierarchy $(M\_\alpha)\_{\alpha\in\omega\_1}$ with $M\_\alpha\cap ON=\alpha$, we can ask whether uncountably many $\alpha$ satisfy "For all sufficiently large $\beta<\omega\_1$, $\alpha$ is parameter-free definable in $M\_\beta$." Now, we can trivially construct examples where the answer is "yes": namely, let $M\_\alpha$ be $L\_{\eta\_\alpha}$ where $\eta\_\alpha$ is the $\alpha$th insightful ordinal; so the general existence question isn't interesting. However, I would be curious if there are any "natural" hierarchies of $M\_\alpha$s which do satisfy this property; especially if their union is $H\_{\omega\_1}$ in some $V$ which is very far from $L$.



Memorable ordinals
It is a very nice question, but unfortunately, the answer is no. 


**Theorem.** There are only countably many memorable ordinals.


**Proof.** Let $\delta$ be a countable ordinal with $L\_\delta\prec
L\_{\omega\_1}$. I claim that every memorable ordinal is less than
$\delta$.


To see this, suppose that $\alpha$ is memorable, as witnessed by
$\beta$. Fix an ordinal $\gamma$ above both $\delta$ and $\beta$ with $L\_\gamma\prec
L\_{\omega\_1}$. It follows that $L\_\delta\prec L\_\gamma\prec
L\_{\omega\_1}$. But since $\beta\leq\gamma$, we know $\alpha$ is
definable in $L\_\gamma$ and hence (since there are no parameters)
definable in $L\_\delta$. So $\alpha<\delta$, as desired. **QED**


