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Gostanian's paper "The next admissible ordinal" (see https://www.sciencedirect.com/science/article/pii/0003484379900251 ), is concerned with the supremum of the $\alpha$-recursive ordinals for various values of $\alpha$; specifically, this supremum is in general smaller than $\alpha^{+}$, the next admissible ordinal after $\alpha$.

In the introduction to this paper, Gostanian mentions a couple of related problems, namely to determine the supremum of the ordinals that are $\Sigma_{n}$ over $L_{\alpha}$, arithmetical over $L_{\alpha}$, $\alpha-\Delta_{1}^{1}$ and whether they are equal to $\alpha^+$ and claims that "some of these questions have interesting answers" that "will be discussed in a forthcoming paper".

I have been unable to find this paper. Has it been written? If not, are the answers to the above questions still known and if so, what are they?

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    $\begingroup$ I suggest adding a higher level tag for more visibility. For example, possible suggestions for a topic like this are "lo.logic" or "computability-theory". $\endgroup$
    – SSequence
    Commented Oct 3, 2019 at 4:14
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    $\begingroup$ Also very simple question (out of curiosity): How often does it happen that $\omega^{CK}_{\alpha}$ is bad but there is a good ordinal ordinal $x$ such that $\omega^{CK}_{\alpha}<x<\omega^{CK}_{\alpha+1}$. For elementary reasons, as I understand, this should definitely happen quite often for small countable $\omega^{CK}_{\alpha}$ [e.g. when clocking positions below are exactly the same]. But how often does this happen in general for larger ordinals? Sorry if this is too off-topic or trivial. $\endgroup$
    – SSequence
    Commented Oct 3, 2019 at 4:19
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    $\begingroup$ Thanks for your suggestion, which I implemented. Concerning your question, I don't know the answer, but I will give it some thought, it is quite interesting. $\endgroup$
    – M Carl
    Commented Oct 7, 2019 at 7:09
  • $\begingroup$ @SSequence I think that all such ordinals necessarily have $\vert\alpha\textrm{-arithmetical}\vert<\alpha$. If we have an $\alpha$-arithmetical well-ordering $R$ of $\alpha$, I would expect that $R$ is $\Sigma_1$-definable over $L_\beta$ for $\beta>\alpha$, including $\beta=x$. Then in order to phrase things in terms of $x$-recursive w.o.s of $x$, we can extend $R$ to a w.o. of $x$ by adding the elements of $x\setminus\alpha$ at the beginning, and by $x$'s goodness the resulting order type remains $<\alpha^+$ - so the o.t. of $R$ was $<\alpha^+$. $\endgroup$
    – C7X
    Commented Apr 24, 2023 at 8:01
  • $\begingroup$ I also believe I've proved these "arithmetically bad" ordinals do not exist below the ordinal of ramified analysis (the least $\beta$ where $L_\beta$ models ZFC-Powerset), mainly by extending some results from "The Next Admissible Ordinal" and a Harrison-like construction that Noah Schweber kindly gave in a comment on this answer. $\endgroup$
    – C7X
    Commented Apr 24, 2023 at 8:03

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