Define "$\alpha$ starts a gap of order $n+1$ and length $\beta$" iff $\mathcal P^n(\omega)\cap (L\_{\alpha+\beta}\setminus L\_\alpha)=\emptyset\land\forall\gamma\in\alpha: L\_\alpha\setminus L\_\gamma\neq\emptyset$ where $\mathcal P^n$ is the powerset operation iterated $n$ times.


Define "$\alpha$ is in a gap of order $n$" as $\forall m<n:\mathcal P^m(\omega)\cap (L\_{\alpha+1}\setminus L\_\alpha)=\emptyset$


Define $\text{ZFC}^-$ as $\text{ZFC}-$(Powerset axiom). This theory is equiconsistent with $Z\_2$, second order arithmetic.
I believe gaps in the constructible universe were first talked about by Putnam in 1963, or at least that's the oldest source I've read, but that's besides the point. [According to Marek and Srebrny](https://pdf.sciencedirectassets.com/272681/1-s2.0-S0003484300X0006X/1-s2.0-0003484374900059/main.pdf?X-Amz-Security-Token=IQoJb3JpZ2luX2VjENj%2F%2F%2F%2F%2F%2F%2F%2F%2F%2FwEaCXVzLWVhc3QtMSJIMEYCIQCxo9GqSQOYZSIIuUy2DSaSS8MTj%2BKHq50tH95EUPNm0AIhAL5fp%2B2gW6Yq05kNaTP4PZoHTM3UhTT9CSA7L8COg%2BrwKvoDCGEQBBoMMDU5MDAzNTQ2ODY1Igx4J6ktYLU7RXb%2BsmYq1wOcQf%2F44N0cNoOBXXPgsLlTVcp4c78i1oFBjZSbP3wuwq6AA2VrkN21XS0EwQGSRmNXfu8kv0RrtnmfbOiAek7fYbLQVMK0rXANYj8SBoSj6LufLnuJrbxf6SvUHmmaZddRgfmK2mQaB7bU%2FWNtkRyTK6021D3LhyJpRo1DNkSRHmoHhSX5QI0s8%2Fv8SZoJS3gtrqrxQ2%2FQc82TYNR6AJS6ZuO6f9Yqs3UfhVItG4H85ZFyLlW9kP4fxwK1j8jZ%2FiTUZKI29HqUuT%2FRlvvw2mP%2FEU%2BQvhbByTJg2xWs6S%2FOrBvfygk7M8sFFGbHC%2FQ6fInUjqrL17jZbiGCUtn90syE1tgqoSc4L7Ao1GmjYiwFyBeMzUFXqkgE%2FTwoh4su%2F%2FsmnygvXoFqQZ5HbywGgtruC8ZdpklpYgmSTJybPcBB%2FklduYZmEo3Oy1QMAqCIbc3WhDgeb9%2BdUVWRwU4lT4Ih1zFCcs4WBxuvOVbMelGl5UStESBgLrI792DK%2FDemC1EyHptDhng3rZFrgz3wOCRtD8v8hxRtEzsCf2Nxi1NhybARf9Ti%2BFyAiUteaXsJA0DSbZa%2Fmrn5TgPitudIIqjVnlzbkYrrAOIhV6DmaA7%2FrQfSgKFz5pIwxKzykQY6pAFbFAKhgacu9s38EN4BA6dk%2BhEUSKV51S6cIi8WURUb0mqKK%2Bkld1ITcJo3%2Fi2ASeCPX1bBrD%2B6BAgT9fEOY9LsJmjXcTD2Boa%2Fv81ZTcY3Hzo411hd6ZsrVDykqwg7pN8FAoFsdXvV7roOMxMIP3K5%2F918%2BNG0k6RSiiv75cHcT4YCkKKJZ%2F%2FK7S%2FyO4c5gdBr14EjqnS9lUjDuwQm8D4ClcBdww%3D%3D&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Date=20220324T164913Z&X-Amz-SignedHeaders=host&X-Amz-Expires=300&X-Amz-Credential=ASIAQ3PHCVTYYYD5USHB%2F20220324%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Signature=29e8ea0c118a303fabdbdd97302ae500b418eacc5ad81ea4382c4ec57ea02b53&hash=a680b28c2b25f6f51d588c3dae657f44b0461067414a3e6e8b15000831d8178b&host=68042c943591013ac2b2430a89b270f6af2c76d8dfd086a07176afe7c76c2c61&pii=0003484374900059&tid=spdf-202fd20b-5df8-4322-af50-0a130923fc8c&sid=056b19b77010d543ae1a4e94453ba94f2960gxrqb&type=client&ua=530152570058545e5d5d&rr=6f10e618297b9be2), the least ordinal that starts a gap of second order is the minimal model height of $\text{ZFC}^-$ and [on p. 372 theorem 3.7] the first ordinal to start a gap of third order is also the minimal model height of $\text{ZFC}^-+\exists\omega\_1$.


I have four questions associated with this property:


1. Does the pattern hold that the least ordinal to start a gap of order $n+2$ is also the minimal model height of $\text{ZFC}^-+\exists\omega\_n$ or $Z\_{n+2}$?
2. If so, are all the ordinals $\alpha$ that start gaps of order $n+2$ exactly those for which
$L\_\alpha\models\text{ZFC}^-+\exists\omega\_n$ and $L\_\alpha\cap\mathcal P^{n+1}(\omega)\models Z\_{n+2}$? If the answer to (1) is unknown then how about (2) when $n=0$?
3. Does the ordinal $\sup\{\min\{\alpha~|~\mathcal P^n(\omega)\cap (L\_{\alpha+1}\setminus L\_\alpha)=\emptyset\}: n\in\mathbb N\}$ itself start any gaps?
4. Is there a countable ordinal that starts a gap of every order (of order $n\forall n\in\mathbb N$) simultaneously?


EDIT: As pointed out by Farmer S, question (4) does not make sense as it is contradicted by page 371 of the very same paper.
Thus, I will change it to the following:


Can an ordinal **be in a gap** of every order (order $n~\forall n\in\mathbb N$) simulteneously?



Can countable ordinals start gaps of every order in the constructible universe?
I presume that by "starts a gap (of order $n$)" you mean "starts a gap of positive length (of order $n$)", since length $0$ is trivial.


Note that the answers given by Monroe Eskew and Fedor Pakhomov were written when there was a different definition of "starts a gap" than what is now there.


1. Yes (by $L\_\alpha\models\mathrm{ZFC}^-$, there are no bounded subsets of $\alpha$ in $L\_{\alpha+1}\backslash L\_\alpha$; and if $L\_\alpha\models$"$\omega\_n$ is the largest cardinal" but $L\_\alpha\not\models\mathrm{ZFC}^-$, then $L\_{\alpha+1}\backslash L\_\alpha$ contains some bounded subset of $\alpha$, hence some subset of its $\omega\_n$).
2. No; you need to add the condition that $L\_\alpha\models$"$\omega\_n$ is the largest cardinal", and then you get the characterization.
3. No (again assuming that the length of a gap has to be $>0$);
let $\alpha$ be that sup.
Then the sequence $\left<\alpha\_n\right>\_{n<\omega}$, where $\alpha\_n$ is the first start of a gap of order $n$, is $\Sigma\_1$-definable over $L\_\alpha$ without parameters, and in $L\_\alpha$, every set is countable. But then it follows that $L\_\alpha$ is the $\Sigma\_1$-hull of the empty set in $L\_\alpha$ (that is, every element of $L\_\alpha$ is $\Sigma\_1$-definable over $L\_\alpha$ from no parameters), which implies that $\alpha$ does not start a gap.
4. No (ignoring the fact that "starts a gap of order $0$" was not defined);
$\alpha$ can not simultaneously start a gap of two distinct orders (this is alluded to in the case of orders 2 and 3 on p. 371 of the paper, near the bottom of the page). E.g. for orders 2 and 3, if cofinally many $\gamma<\alpha$ project to $\omega$, then $\omega$ is the largest cardinal in $L\_\alpha$. (Note Monroe's comment was written when the definition of "starts a gap" was different.)


