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, 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:

- 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 $Z_{n+2}$?
- 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 $n=0$?
- 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?
- 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?

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