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sniffnoyOops next day: Not ordinary exponentiation. See below.
I've posted this whole mess as a question to MathOverflow. After all, I don't even really know surreals, I'm clearly not equipped to answer it! Nor do I have the time; I still have to finish revising the first of these integer complexity papers, except of course not till Tuesday because first I have a problem set for Schubert calculus.
But for those too lazy to click through the link... things are bad.
Note: To avoid confusion between ωx in the usual sense and ωx in the sense of the exponentiation under discussion, I'll write the former as ω^x and reserve superscripts for the latter.
Recall that Wikipedia's definition (I still have no idea where it's taken from, so WP will just have to take the credit for now) can be extended in one of two ways: By replacing "2" with other numbers, or by doing the usual thing with logs. These don't agree! And of course you could replace 2 with another number and *then* do the usual thing with logs, in your new base, to get an *entirely different definition*. Hooray!
(Why don't these agree? Well, as an example, consider that if we try to generalize by replacing the base, we still get 3ω=ω, while if we try using logs, we get 3ω>ω.)
In fact, the "replace the base" definition seems pretty suspect for another reason: it looks like if we plug in two ordinals, it agrees with ordinary ordinal exponentiation. Now I don't know, maybe ordinary ordinal exponentiation actually works really well with natural addition and natural multiplication. I haven't checked, after all, because I only just noticed this problem. But it doesn't seem very likely.
Oops: Not ordinary exponentation, but rather the analogue of that based on natural multiplication. The exponentiation without a name that Jacobsthal didn't consider...
In fact if we use Gonshor's definition, ωω actually won't be an ordinal at all. Disappointing, but perhaps to be expected -- if surreal exponentiation really did yield a "natural" ordinal exponentiation, I think someone would have noticed in the past 30 years. If we apply Gonshor's definition, and the various theorems he proves about it, we find ωω=exp(ωlogω)=exp(ω*ω^(1/ω))=exp(ω^(1+1/ω))=ω^(ω^(1+1/ω)).
So that's a strike against the idea of "natural exponentiation", unless of course ordinary exponentiation turns out to be suitably "natural". There is Jacobsthal's definition -- or you could do the equivalent but working from natural multiplication rather than his weird multiplication -- but, blech.
I mean, let's say you wanted to try to define it. If you try to define it recursively, powers of 0 will be a problem. If you try to define it "computationally" -- well, you can't raise polynomials to transfinite powers! That leaves the "constructive" approach, which, I don't know, maybe it's possible -- but part of what makes the other operations nice in the first place is the fact that there's multiple ways to think about them, not just the one.
In short, I'm leaving standing the question on MathOverflow (because really guys, why is this a big mess, this should not be a big mess, this should be answered), but there is probably no such thing as "natural exponentiation" (I don't count Jacobsthal's definition).
-Harry
I've posted this whole mess as a question to MathOverflow. After all, I don't even really know surreals, I'm clearly not equipped to answer it! Nor do I have the time; I still have to finish revising the first of these integer complexity papers, except of course not till Tuesday because first I have a problem set for Schubert calculus.
But for those too lazy to click through the link... things are bad.
Note: To avoid confusion between ωx in the usual sense and ωx in the sense of the exponentiation under discussion, I'll write the former as ω^x and reserve superscripts for the latter.
Recall that Wikipedia's definition (I still have no idea where it's taken from, so WP will just have to take the credit for now) can be extended in one of two ways: By replacing "2" with other numbers, or by doing the usual thing with logs. These don't agree! And of course you could replace 2 with another number and *then* do the usual thing with logs, in your new base, to get an *entirely different definition*. Hooray!
(Why don't these agree? Well, as an example, consider that if we try to generalize by replacing the base, we still get 3ω=ω, while if we try using logs, we get 3ω>ω.)
In fact, the "replace the base" definition seems pretty suspect for another reason: it looks like if we plug in two ordinals, it agrees with ordinary ordinal exponentiation. Now I don't know, maybe ordinary ordinal exponentiation actually works really well with natural addition and natural multiplication. I haven't checked, after all, because I only just noticed this problem. But it doesn't seem very likely.
Oops: Not ordinary exponentation, but rather the analogue of that based on natural multiplication. The exponentiation without a name that Jacobsthal didn't consider...
In fact if we use Gonshor's definition, ωω actually won't be an ordinal at all. Disappointing, but perhaps to be expected -- if surreal exponentiation really did yield a "natural" ordinal exponentiation, I think someone would have noticed in the past 30 years. If we apply Gonshor's definition, and the various theorems he proves about it, we find ωω=exp(ωlogω)=exp(ω*ω^(1/ω))=exp(ω^(1+1/ω))=ω^(ω^(1+1/ω)).
So that's a strike against the idea of "natural exponentiation", unless of course ordinary exponentiation turns out to be suitably "natural". There is Jacobsthal's definition -- or you could do the equivalent but working from natural multiplication rather than his weird multiplication -- but, blech.
I mean, let's say you wanted to try to define it. If you try to define it recursively, powers of 0 will be a problem. If you try to define it "computationally" -- well, you can't raise polynomials to transfinite powers! That leaves the "constructive" approach, which, I don't know, maybe it's possible -- but part of what makes the other operations nice in the first place is the fact that there's multiple ways to think about them, not just the one.
In short, I'm leaving standing the question on MathOverflow (because really guys, why is this a big mess, this should not be a big mess, this should be answered), but there is probably no such thing as "natural exponentiation" (I don't count Jacobsthal's definition).
-Harry
