Timeline for Weaker versions of Gandy ordinals
Current License: CC BY-SA 4.0
12 events
| when toggle format | what | by | license | comment | |
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| Jul 7, 2023 at 0:48 | comment | added | C7X | @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 $L_\beta\vDash\textrm{ZFC-Powerset}$". | |
| Apr 24, 2023 at 8:03 | comment | added | C7X | 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. | |
| Apr 24, 2023 at 8:01 | comment | added | C7X | @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 $<\alpha^+$. | |
| S Oct 17, 2019 at 14:01 | history | bounty ended | CommunityBot | ||
| S Oct 17, 2019 at 14:01 | history | notice removed | CommunityBot | ||
| S Oct 9, 2019 at 12:07 | history | bounty started | M Carl | ||
| S Oct 9, 2019 at 12:07 | history | notice added | M Carl | Draw attention | |
| Oct 7, 2019 at 7:09 | comment | added | M Carl | 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. | |
| Oct 7, 2019 at 6:37 | history | edited | M Carl |
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| Oct 3, 2019 at 4:19 | comment | added | SSequence | 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. | |
| Oct 3, 2019 at 4:14 | comment | added | SSequence | 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". | |
| Sep 24, 2019 at 9:25 | history | asked | M Carl | CC BY-SA 4.0 |