Edit 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
