Talk:Nonrecursive ordinal

I will move this page to "unrecursive ordinals"

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Semi-nonprojectible

Is there a source for the name "semi-nonprojectible"? I can't find any results on Google other than this page, and also not on StackExchange C7XWiki (talk) 03:57, 3 September 2021 (UTC)

I added this a while ago and made up the term because I didn't know you needed citations xD I will remove it. Same goes for the Devlin-Jech ordinal. I like to have things named and neat and organised, but I guess it is more important to have valid sources. Binary198 (talk) 09:59, 26 October 2021 (UTC)

Thanks for clarifying! Wikipedia's citation policy is stricter than Googology Wiki's by the way C7XWiki (talk) 21:46, 28 October 2021 (UTC)

Unclear reference to the first uncountable ordinal

What does "$'\omega _{1}exists'$" mean in "The smallest ordinal α such that $L_{\alpha }\models {\mathsf {ZFC^{-}}}+'\omega _{1}exists'$."? Surely, ZFC implies the existence of ω₁ so what are you adding? JRSpriggs (talk) 02:31, 4 September 2021 (UTC)

Here ${\mathsf {ZFC^{-}}}$ means ${\mathsf {ZFC}}$ without the powerset axiom, so we accept "ω₁ exists" as an axiom C7XWiki (talk) 22:35, 25 October 2021 (UTC)

@C7XWiki: Yes, thanks for clarifying that. Perhaps one should consider a stronger axiom, to wit, that every set has a Hartogs number. JRSpriggs (talk) 14:42, 29 November 2022 (UTC)

Unusable reference

To C7XWiki: I do not understand the notation used the section on stable ordinals. I tried to look it up in your reference to the "Spectrum of L". However, my browsers will not allow me to down load the PDF file because they say that it is unsafe. Could you please explain here what $L_{\alpha }\preceq _{1}L_{\alpha +1}$ means. JRSpriggs (talk) 14:42, 29 November 2022 (UTC)

@JRSpriggs $\preceq _{1}$ means $\Sigma _{1}$-elementary sumbodel, $L_{\alpha }\preceq _{1}L_{\alpha +1}$ is actually an abuse of notation for $(L_{\alpha },\in )\preceq _{1}(L_{\alpha +1},\in )$. $A\preceq _{1}B$ means that for any $\Sigma _{1}$ formula $\phi$, for any parameters $a_{0},\ldots ,a_{n}\in A$, we have $(A\vDash \phi (a_{0},\ldots ,a_{n}))\iff (B\vDash \phi (a_{0},\ldots ,a_{n}))$. The unsafe PDF is a strange problem, I haven't gotten this warning and seem to be able to download the PDF without a problem. I can look for a different site hosting the reference, or also cite another paper defining stability (an early one with many important results is Inductive definitions and reflecting properties of admissible ordinals, 1974.) C7XWiki (talk) 06:49, 1 December 2022 (UTC)

Edit: Unfortunately I can't find any other open-access link for Spectrum of L, but I can give a link to its WorldCat and EuDML pages. C7XWiki (talk) 07:03, 1 December 2022 (UTC)

I said I do not understand it because it seems impossible to me. α = λ+n where λ is a limit ordinal and n is finite. So α+1 would have the finite part being one larger than α would. You should be able to construct a Σ₁ formula that would distinguish between them, is it not so? JRSpriggs (talk) 17:06, 1 December 2022 (UTC)

OK, I think I get it now. α is a limit ordinal and the parameters must be constructed before that stage. So there is no way to say that the existentially quantified variable is α or constructed at the last stage in L_α+1. As long as nothing new is being built from below, the equivalence is possible. JRSpriggs (talk) 21:08, 1 December 2022 (UTC)

Now, I found a way to work around the problem I had earlier and download "Spectrum of L". Thanks for your help. JRSpriggs (talk) 23:57, 1 December 2022 (UTC)

@JRSpriggs I think you're correct, there's no formula that 'forces' the existentially quantified variable to be α. If we carried out such a Σ₁ definition of uniquely α within L_α+1, it would reflect down to some set α' in L_α and by upwards absoluteness such α' would satisfy that definition within L_α+1 as well. In RichterAczel74 (at theorem 1.18, the half of the proof starting at "conversely, let α be α+1-stable"), there's a nice proof that when $L_{\alpha }\preceq _{1}L_{\alpha +1}$, $\alpha$ is $\Pi _{n}$-reflecting for all n<ω, which in my opinion makes the machinery behind stability clearer. C7XWiki (talk) 02:25, 2 December 2022 (UTC)

Page Church-Kleene ordinal

I think this page should be re-split into Church-Kleene ordinal, and another page about nonrecursive ordinals. (Maybe large countable ordinal has enough information about these?) There is a quite a bit of information about ω₁^CK, especially publications during the 60s and 70s about meta-recursion theory, that are worth mentioning, and some of these results may not extend to higher admissible ordinals. C7XWiki (talk) 22:08, 21 February 2023 (UTC)

Sounds good to me. 67.198.37.16 (talk) 05:52, 15 April 2024 (UTC)