Do all limit $\alpha \in \omega_1^L$ satisfy $L_\alpha \models V=HC$?

Question

In Gaps in the constructible universe, Marek and Srebrny, 1973 a gap ordinal and the start of a gap are defined as follows

$\alpha$ is a gap ordinal iff $(L_{\alpha+1}-L_\alpha)\bigcap \mathcal{P}(\omega) = \emptyset$

$\alpha$ starts a gap iff it is a gap ordinal and $\forall\beta\in\alpha((L_\alpha-L_\beta)\bigcap \mathcal{P}(\omega) \neq \emptyset)$

They present the following Lemma and Corollary, the Corollary without any proof or comment:

Lemma 2.4. If $\alpha$ starts a gap, then it is a limit ordinal.

Corollary 2.4. If $\alpha$ starts a gap, then $L_\alpha \models V=HC$.

This presentation of course makes it seem as if every limit $\alpha \in \omega_1^L$ satisfies $L_\alpha \models V=HC$, but I haven't been able to prove this fact, and am not sure it's true.

I've also tried proving the Corollary by additionally using the fact that $\alpha$ starts a gap as follows:

As per the definition, this means there are in $\alpha$ arbitrarily big non-gap ordinals. By a result of Boolos mentioned in the article, if $\beta$ is a non-gap ordinal, there is an $E_\beta \in L_{\beta+1}$ such that $\langle Field(E_\beta), E_\beta \rangle \cong \langle L_\beta, \in \rangle$, where $Field(E_\beta) \subseteq \omega$. Thus, given an $x \in L_\alpha$, it'll belong to some $L_\beta$ such that there's an $E_\beta \in L_\alpha$, and thus $L_\alpha$ will prove $x$ countable, given the isomorphism between $L_\beta$ and $Field(E_\beta)$ also belongs to $L_\alpha$. But I'm not sure of this last fact. Checking the proof of Theorem 1 in Degrees of unsolvability of constructible sets of integers, Boolos and Putnam, 1968, the fact will be true if the Mostowski collapse is always definable in the constructible hierarchy in $\omega$ steps. More concretely, if when $S \in L_\beta$, then the function $mos: S \leftrightarrow mos(S)$ belongs to $L_\alpha$.

Answer 1

As Asaf mentioned, this is not true. It's indeed true that when $\alpha$ is a gap ordinal $L_\alpha\vDash\textrm{V=HC}$, but when considering some ordinals above gap ordinals we get points where it fails. For example, this example uses a result from Arai's "A sneak preview of proof theory of ordinals" (p.17): let $\alpha$ start a gap of length $\alpha^+$ (where $\xi^+$ denotes the least admissible $>\xi$.) Then not only does $L_{\alpha^+}\vDash\lnot(\textrm{V=HC})$, but also $\omega_1^{L_{\alpha^+}}$ (i.e. what $L_{\alpha^+}$ thinks is $\omega_1$) is $\alpha$.

We can find larger cardinals $\kappa$ where this premise fails for $\textrm{V=H}_\kappa$ as well - the minimal $L_\alpha$ modelling ZFC satisfies powerset so it believes all sorts of accessible cardinals exist, but $\alpha$ is countable according to David Madore's "A Zoo of Ordinals" (p.6).