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Background I am not a professional mathematician. I am researching Surreal numbers & games for fun (I think they are truly beautiful). If this question is not appropriate here, I beg forgiveness & ask that it please be migrated it to MSE.

I read More Infinite Games by John H Conway recently & have not been able to stop thinking about the following line (page 4):

Note that $\uparrow$ is not a number: it is the value of a game, which is a more subtle concept. Also note that $\frac{1}{\uparrow}$ is not defined since it would be bigger than all surreal numbers and there are no such numbers. (In fact, it does exist but is one of the Oneiric numbers.)

Unfortnuately, the only thing on the internet about the Oneiric numbers is this paper. If my current understanding is correct, a part of the problem is that the reciprocal of games is not (yet) defined. As previously stated, I'm not an expert & have no clue if working out the reciprocals of games is an impossible (or just very difficult) task, or even if it has already been done.

It has been suggested by a couple people that I try to find someone who knew Conway & might have some idea of what he was thinking. I would be elated to do so. Alternatively, perhaps someone who is knowledgable about the Surreals might venture to take a crack at a definition.

Thanks for taking the time to read this question, I hope it isn't a waste of anyone's time.

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    $\begingroup$ As a sidenote, the adjective "oneiric" (of or pertaining to dreams) is one of the most beautiful adjectives I've seen in front of a mathematical object, and perfectly captures what should come after "surreal". $\endgroup$
    – Carl-Fredrik Nyberg Brodda
    Commented May 13, 2020 at 17:41
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    $\begingroup$ Naively, I would expect Conway's remark to mean that there is a way of treating $1/\uparrow$ as a symbol in a formal manner, admitting all the usual operations except those which would cause the entire building to collapse, and that the resulting calculus is what he means by Oneiric numbers (which I agree is an engaging name). Is there any reason to suppose that he had something bigger in mind? Sometimes a remark such as this (especially when a well-chosen neologism is involved) can be so suggestive that it leads one to expect grand things, while the reality is much more prosaic. $\endgroup$
    – R.P.
    Commented May 13, 2020 at 22:33
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    $\begingroup$ @RP_ I don't disagree, but division (arguably even multiplication) on games that aren't numbers or stars typically "causes the entire building to collapse", so I am curious as to what justification there is for writing $1/\uparrow$. $\endgroup$
    – Mark S.
    Commented May 14, 2020 at 1:24
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    $\begingroup$ It wouldn't surprise me if this were one of Conway's jokes; i.e., such a thing exists only in our dreams, or we can dream that one day someone might make sense of it. Note for example that Conway would sometimes say that something was "cohomology" when it wasn't literally cohomology; it was one of his jokes. $\endgroup$
    – Timothy Chow
    Commented May 14, 2020 at 17:03
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    $\begingroup$ The only thought I have to add is that Oneiric numbers almost certainly involve proper classes. Note that "Oneiric" is capitalized, suggesting "On", the class of ordinals. Perhaps they even involve classes of classes, or higher iterations. $\endgroup$
    – Tom Kerruish
    Commented Jun 9, 2021 at 8:12

1 Answer 1

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I emailed John Conway about this very thing over ten years ago. His response (paraphrasing) was along the lines of RP’s comment; if you treat 1/up as a formal entity nothing breaks, but there wasn’t more to the theory than that. I do not have an account, and I don’t have the email still, so I neither want nor expect the bounty. One other detail I remember is that he represented 1/up with the Roman numeral I.

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    $\begingroup$ If no other answer comes along (and if one does, I doubt it will top this one), you will be awarded the bounty anyway. Use this account as you please, but it would be nice to know more of who this correspondent of Conway's (meaning you) is. Gerhard "Or We Can Keep Guessing" Paseman, 2020.05.16. $\endgroup$
    – Gerhard Paseman
    Commented May 16, 2020 at 21:44
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    $\begingroup$ I’m hardly a correspondent; I emailed him exactly once, and it was to ask this question. In hindsight, I’m kind of surprised he responded to a shallow email from a college student he’d never heard of. $\endgroup$
    – Jeffrey
    Commented May 16, 2020 at 22:44

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