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If addition is the first hyperoperation, multiplication is the second, and the $(\alpha+1)$th hyperoperation is repeated occurrences of the $\alpha$th one. Is it possible for a limit ordinal (for example $\omega$) to be $\alpha$ and we use an nth term in its fundamental sequence as the $\alpha$. I don’t know if that’s made any sense so here’s an example.

If we take $\cdot^\alpha$ to mean the $\alpha$th hyperoperator, could we evaluate $n\cdot^\omega m$ to $n\cdot^n m$. This wouldn’t be very unusual so I am going to assume so. Now my question is:

Is there a way to evaluate a $\omega_1^{CK}$th hyperoperator equation into something we could turn into a finite number.

I’m assuming not as the Church-Kleene ordinal has no fundamental sequence that I know of, but there may be something that I haven’t thought of.

Logan M
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  • Is there a reason we cannot evaluate $\alpha$th hyperoperations for $\alpha\ge\omega_1^{CK}$? It seems well-defined and we can evaluate some values. I guess the real problem is it could not be recursive. – Hanul Jeon Nov 11 '18 at 05:57
  • That’s the reason I decidedly to go for the Chirch-Kleene ordinal, because if you could do it fro the smallest non-recursive ordinal, it follows you could do it for the others. But no, I don’t see any reason against $\alpha$ being a greater ordinal, the problem is their fundamental sequences can’t be anything recursive and that’s makes it tricky. – Logan M Nov 11 '18 at 08:46
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    I downvoted this question because you can't jump from "What is $\omega+\omega^2$, because if it is not $\omega^2$ I can prove CH" into "How do I define an $\omega_1^{CK}$th hyperoperation" in a six hours window of time. – Asaf Karagila Nov 11 '18 at 09:07

1 Answers1

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There's an interesting situation here: while the question itself isn't a duplicate, a previous answer of mine I think resolves it. I don't quite know what to do in this case, so I've written a same-spirit, different-phrasing below, and not marked this question as a duplicate; to avoid "double-dipping" on reputation, I've marked my answer community wiki.

The difficulty here is that limit stages wind up more complicated than they look at first. The weasel words here are

its fundamental sequence.

The key point is, there is no unique fundamental sequence for a limit ordinal! The "$\lambda$th hyperfunction" isn't fully determined by $\lambda$; we also have to choose a fundamental sequence for $\lambda$ (and indeed for each limit ordinal $<\lambda$). One might hope that this choice winds up being insignificant, but in fact it's not: different hyperoperations corresponding to the same ordinal can vary wildly.

If you don't require canonicity, then there's no problem - fixing a system of fundamental sequences for all limit ordinals up to a certain point will define a hyperoperation at that point - but the point is that no good "all-the-way-up" (or even a-little-bit-of-the-way-up) theory of hyperoperations exists.

So what happens when we actually do talk about ordinal-indexed hyperoperations?

When we say something like "the $\omega^\omega$th hyperoperation" - e.g. in the context of fast-growing hierarchies - in the background we're assuming a fixed notation system for the ordinals in question, and this amounts to a choice of fundamental sequences. But this notation system can only reach a small ways up (e.g. to $\epsilon_0$ in the case of the Wainer hierarchy); in particular, there is no recursive way to assign a unique fundamental sequence to every recursive limit ordinal.

  • This is reflected in the non-linear structure of Kleene's $\mathcal{O}$: Kleene provides a system to (essentially) assign fundamental sequences to recursive presentations of recursive ordinals, but two different presentations of the same ordinal can get different wildly fundamental sequences.

That said, there is a arguably-reasonable way one can try to go beyond $\omega_1^{CK}$ (indeed, all the way up to $\omega_1^L$, the first $L$-uncountable ordinal) in a canonical way - mastercodes. But this gets extremely technical. And it's consistent that $\omega_1^L$ itself is countable, so even this trick won't get us very far.

Noah Schweber
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  • There is a paper by the name "An extended arithmetic of ordinal numbers", which seems at least somewhat accessible (even to me). I haven't really read it or even skimmed it though. But the introductory paragraph seems to be of relevance to idea of hyperoperations. Anyway, I thought I would mention it ...... partly because there seem to a good number of questions under this tag that have at least some link to "hyperoperations", but I haven't this particular paper mentioned (at least for the questions that I have viewed). Maybe I missed it or maybe there is some specific reason for it. – SSequence Nov 13 '18 at 18:46
  • The dead link to the paper on mastercodes used to go to this paper: Harold T. Hodes, Jumping Through the Transfinite: The Master Code Hierarchy of Turing Degrees, The Journal of Symbolic Logic, Vol. 45, No. 2 (Jun., 1980), pp. 204-220; DOI: 10.2307/2273183, JSTOR – Martin Sleziak May 14 '22 at 05:59