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sniffnoyEdit again Feb 20: Added note about Wikipedia's definition and ordinals.
Edit: Added small note about how one could verify equivalence with Kruskal's definition. Also, added a note to this entry pointing out that the possible definition of "natural exponentiation" I suggest therein fails very badly.
So having looked at Jacobsthal's paper I think his "natural exponentiation" does not deserve the name. At least, if this is the right paper and the right operation. It's called "Zur Arithmetik der transfiniten Zahlen". In it, he considers natural addition, and then uses it to define a new multiplication and exponentiation operation on the ordinals, so I figure this must be it, right?
But the multiplication he defines is *not* natural multiplication. Rather, he uses the recursion for ordinary multiplication, using natural addition in place of ordinary addition. This is a new multiplication based on natural addition alright, but it's not natural multiplication (consider that under this multiplication, 2ω=ω, for the same reason it does with ordinary multiplication).
Then he defines a new exponentiation the same way (usual recursion, but based on his new multiplication). I think it's pretty clear why this can't sensibly be called "natural exponentiation".
But there does seem to be such a thing as surreal exponentiation. Or, let's for now at least assume there really is such a thing. (I mean, certainly there exist definitions of it, the question is whether they're all the same.) Then if we had ordinals x and y, we could take xy in the surreals... and if the result is always another ordinal, perhaps we could call that natural exponentiation? It would have the right algebraic properties, though that leaves the question of how it could be defined in a purely ordinal manner (recursively? constructively? computationally?). Question is, for ordinals x and y, is xy always another ordinal in the first place? I have no idea.
Edit: If we use Wikipedia's definition of 2x and generalize it by allowing other ordinals in place of 2, it would appear that for ordinals x and y, xy is indeed again an ordinal.
So. Surreal exponentiation. Last entry I mentioned there was a definition due to Kruskal. I should note that I have never seen this definition; On Numbers and Games, and Gonshor's book, both mention it, but neither state it. They also both mention that it hasn't been published, and, well, it looks like it still hasn't.
But Gonshor's book does suggest that his definition is equivalent to Kruskal's older one. Annoyingly, it doesn't explicitly state it. So I gather it's probably equivalent, but perhaps I should still see if there's someone I can ask about it. (But who would have seen Kruskal's definition? Well, I guess there's enough people on MathOverflow for there to be a chance someone has.) Meanwhile, I'm not too certain what Alling is doing -- remember, I don't really know surreal numbers, I was just looking this up for fun! -- but it looks like it's probably the same as Gonshor's, except that he only defines it for non-infinite surreals? I'm not too sure.
Addendum: Actually, despite Kruskal's work not being published, there is a way one could verify equivalence with it (though I'm certainly not going to). Conway mentions that Kruskal's definition of exp, was inverse to a certain definition of log; and it looks like he gives enough information to determine what that definition of log was.
This leaves the question of where the hell Wikipedia's uncited definition of 2x comes from. With any luck maybe that'll turn out to be Kruskal's original definition. :P Still, at least I have reason to believe that there's at most 2 definitions and not, say, four of them. And hey, there's a good chance those remaining two are the same, right? :-/
-Harry
Edit: Added small note about how one could verify equivalence with Kruskal's definition. Also, added a note to this entry pointing out that the possible definition of "natural exponentiation" I suggest therein fails very badly.
So having looked at Jacobsthal's paper I think his "natural exponentiation" does not deserve the name. At least, if this is the right paper and the right operation. It's called "Zur Arithmetik der transfiniten Zahlen". In it, he considers natural addition, and then uses it to define a new multiplication and exponentiation operation on the ordinals, so I figure this must be it, right?
But the multiplication he defines is *not* natural multiplication. Rather, he uses the recursion for ordinary multiplication, using natural addition in place of ordinary addition. This is a new multiplication based on natural addition alright, but it's not natural multiplication (consider that under this multiplication, 2ω=ω, for the same reason it does with ordinary multiplication).
Then he defines a new exponentiation the same way (usual recursion, but based on his new multiplication). I think it's pretty clear why this can't sensibly be called "natural exponentiation".
But there does seem to be such a thing as surreal exponentiation. Or, let's for now at least assume there really is such a thing. (I mean, certainly there exist definitions of it, the question is whether they're all the same.) Then if we had ordinals x and y, we could take xy in the surreals... and if the result is always another ordinal, perhaps we could call that natural exponentiation? It would have the right algebraic properties, though that leaves the question of how it could be defined in a purely ordinal manner (recursively? constructively? computationally?). Question is, for ordinals x and y, is xy always another ordinal in the first place? I have no idea.
Edit: If we use Wikipedia's definition of 2x and generalize it by allowing other ordinals in place of 2, it would appear that for ordinals x and y, xy is indeed again an ordinal.
So. Surreal exponentiation. Last entry I mentioned there was a definition due to Kruskal. I should note that I have never seen this definition; On Numbers and Games, and Gonshor's book, both mention it, but neither state it. They also both mention that it hasn't been published, and, well, it looks like it still hasn't.
But Gonshor's book does suggest that his definition is equivalent to Kruskal's older one. Annoyingly, it doesn't explicitly state it. So I gather it's probably equivalent, but perhaps I should still see if there's someone I can ask about it. (But who would have seen Kruskal's definition? Well, I guess there's enough people on MathOverflow for there to be a chance someone has.) Meanwhile, I'm not too certain what Alling is doing -- remember, I don't really know surreal numbers, I was just looking this up for fun! -- but it looks like it's probably the same as Gonshor's, except that he only defines it for non-infinite surreals? I'm not too sure.
Addendum: Actually, despite Kruskal's work not being published, there is a way one could verify equivalence with it (though I'm certainly not going to). Conway mentions that Kruskal's definition of exp, was inverse to a certain definition of log; and it looks like he gives enough information to determine what that definition of log was.
This leaves the question of where the hell Wikipedia's uncited definition of 2x comes from. With any luck maybe that'll turn out to be Kruskal's original definition. :P Still, at least I have reason to believe that there's at most 2 definitions and not, say, four of them. And hey, there's a good chance those remaining two are the same, right? :-/
-Harry
no subject
Date: 2012-12-24 10:16 pm (UTC)When he calls it 'natural' exponentiation, he's actually describing the recursive nature of the definition, not its intuitiveness or sensicality. One of the really cool things about the surreals is that one or two rules can result in the largest ordered set that can possibly be constructed, and another two rules makes it the largest ordered field (addition and multiplication). People are looking for 'natural rules' that can be applied iteratively such that the results are both as desired and consistent (there have been attempts to define derivatives in this way, in fact). His definition of exponentiation is natural in that it is a single rule which defines the entirety of exponentiation when used iteratively.
As for Kruskal! Kruskal didn't actually HAVE a written definition, which is one of the reasons you had trouble finding it. He left 'hints' specifically for Gonshor to figure out. When Gonshor says his is equivalent, he's saying what he wrote down is equivalent to what Kruskal was pushing him towards. That's why Kruskal is credited with the equation, having done all of the creative work, but Gonshor is referenced. Both were/are quite brilliant though! Oh, and don't misunderstand Conway: when he's talking about inverses, it turns out that ln is defined to be the inverse of the exponential function. Then, something like 2^x is defined as e^ln(2^x). Which, you may gather, is an entirely useless statement if you want to actually figure out something like (1/w)^(3w^7). To the best of my knowledge, there is no surreal definition of log which isn't defined as the inverse of the exponential. The function, while natural and 'capable' of providing an answer for any exponent and base, is quite unhelpful.
no subject
Date: 2012-12-26 06:08 am (UTC)You'll see I wrote a bit more about this over the next few entries. But to summarize:
I eventually ended up asking about this on MathOverflow here. Philip Ehrlich gave essentially the same answer you gave.
I still have zero idea where the definition of 2^x that was on Wikipedia at the time came from (it doesn't agree with the definition you and everyone else give, IIRC). And so I eventually removed it from the entry for being unsourced. If you want, here's the version of the article from before I removed that. Tracing through the history, it was apparently added by one Michael K. Edwards, who is impossible to contact. So, I dunno.
As for Jacobsthal's multiplication and exponentiation, and related operations, well, I played around with that a bit more and wrote about it here. (Or originally here, if you want to comment.) I don't suppose you have any idea whether that last (bolded) relation is new or not, do you? It seems like the sort of thing someone ought to have come up with before. Honestly I don't know what I'd do with it if it were -- dunno how I'd get a whole paper out of it. This isn't really my specialty, as you can probably tell; I just like playing around with infinities. (Though funnily enough ordinal arithmetic and natural operations have turned out to be relevant to what I'm working on anyway, though that's only with ordinals less than ωω...)
(Also, your explanation of why Jacobsthal called his exponentiation "natural" seems a bit fishy -- it seems to me that could just as well describe ordinary exponentiation. Or were you getting Jacobsthal's exponentiation mixed up with exponentiation in the surreals?)
Anyway, thanks for commenting! I don't suppose you have some handle we can refer to you by should you comment further?