On Saturday 17 November 2012 15:50:47 Andy Latto wrote: [me, about surreal numbers:]
I think that if you reinterpret .9 + .09 + .009 + ... as a sum over *all the ordinals* then we can rightly say that it converges to 1.
I'm very skeptical that one could get a consistent definition for sums over classes that are that large. Even the first step of defining raising 10 to ordinal powers seems problematic.
Maybe you're right -- though there's a note on p38 of my edition of ONAG saying that Simon Norton invented a way of doing integration that lets you define logarithms and hence arbitrary (?) powers x^y. But it doesn't give any details).
But what if it's just a sum over the finite integers? For any infinitesimal h, all the finite partial sums are < 1-h; so it certainly shouldn't converge to 1; it can't converge to any ordinary real number < 1 because some of the partial sums are bigger; if it converges to 1-h for some infinitesimal h, why not 1-2h or 1-h/2?
And again, you have to deal with the geometric series proof. If .99999 is 1-h, wouldn't 9.99999 be both 10 - 10h (because you multiplied each term by 10) and 10 - h (because you added 9 to 1-h), so h = 0.
I repeat: it can't converge to 1 if "converge" means anything like what it means when dealing with ordinary real numbers, because there are lots of upper bounds for the series that are strictly less than 1.
Well, if h is 1/omega then maybe 1-h is the *simplest* thing satisfying some convenient property.
Then the same simplicity would presumably give 9.99999.... = 10 - 1/omega, so you'd have to give up the "multiplication by a scalar" property in exchange for some as-yet-unspecified simplicity property. That could work.
Right. (I thought I'd kinda said that, but on rereading what I wrote I see that I failed to. Sorry.) * On looking a bit further in ONAG I find the following proposal: write numbers as sums of the form sum over all Numbers y of r(y) omega^y where r(y) vanishes except for y in some descending sequence indexed by an ordinal. Then say that a series converges if (1) there's some single ordinal-indexed descending sequence in which all the nonzero exponents in the series lie, and (2) for each exponent, the sums of the corresponding r(y) converge. In the special case of series of real numbers, this is exactly, and rather boringly, the same as saying that the series converges in the usual sense in R. (Because then all your coefficients are zero except for those of omega^0.) This all seems rather unsatisfactory to me, but I haven't anything more convincing to offer. -- g