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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?

M Carl
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    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". – SSequence Oct 03 '19 at 04:14
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    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. – SSequence Oct 03 '19 at 04:19
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    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. – M Carl Oct 07 '19 at 07:09
  • @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 $<\alpha^+$. – C7X Apr 24 '23 at 08:01
  • 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. – C7X Apr 24 '23 at 08:03
  • @MCarl (I didn't think this comment was enough for an answer) Only 4 publications by Gostanian appear on Scopus, plus two more unique one on PhilPapers and EuDML respectively. Google Scholar only contains new citations, none of which look like the sequel. There are two results in Aguilera's "Effective Cardinals and Determinacy in Third-Order Arithmetic" which may be helpful in recovering Gostanian's work, lemma 9 (in view of lemma 5), plus lemma 11 in order to lower-bound the non-2-Gandy ordinals by "least $\beta$ such that $L_\beta\vDash\textrm{ZFC-Powerset}$". – C7X Jul 07 '23 at 00:48

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