On 11/17/2012 6:25 AM, Gareth McCaughan wrote:
On Saturday 17 November 2012 05:10:23 Andy Latto wrote:
But I don't understand why you think the proof that .9999... = 1 requires continuity. [...] Then all we need to assume about such sums to prove that if it has a sum, that sum must be 1 is exactly the three properties you describe below:
If m in R and b_i = m a_i for all i in Z, then sum(b) = m sum(a). If c_i = a_i + b_i for all i in Z, then sum(c) = sum(a) + sum(b).
We would want sums to be preserved by shifting the sequence left or right,
e.g: If k in Z and b_i = a_{i+k} for all i in Z, then sum(b) = sum(a).
The second of those properties doesn't hold even in R without the proviso that all three sums converge; otherwise you'd have 1-1+1-1+1-1... being zero and similarly for -1+1-1+1-1+1..., and with those in hand plus the principle that a0 + sum{1..oo}ai = sum{0..oo}ai you can prove 0=1. So perhaps the right question is: is there actually any such thing as the value of .9 + .09 + .009 + ... in the surreal numbers? The above snippet was not in regard to surreal numbers, it was in regard to possibly assigning a consistent real value to double-ended infinite sums. I don't understand surreals well enough to remark intelligently on them, except to note that in that setting, it is perhaps possible or even reasonable to define 0.9999... in such a way that we could shoehorn an infinitesimal between 0.9999... and 1, thus providing a reasonable setting in which we might conclude 0.9999... =/= 1.
With regard to the double-ended sums, I was hoping to treat them as a formalism, and avoid issues of convergence (which end do you converge toward?). Thus we would assign a reasonable value to some class of basic expressions (e.g, ... + 0 + 0 + 0 + 9 + 0 + 0 + 0 + ... = 9), and apply some formal arithmetic rules (value preserved by shifting, term-by-term sum, term-by-term scalar multiplication) and see where that leads us (I expect to a contradiction). But supposing a consistent theory of formal double-ended sums can be developed this way, by starting with base class sums (the values we decree for certain simple double-ended sums) and geometric double-ended sums (which always sum to 0 according to the arithmetic formalisms), we might construct consistent sums for a more general class of double-ended sums. Then by embedding standard right-infinite sums by left zero padding, we might arrive at a way to consistently evaluate left-infinite sums and perhaps even some divergent sums (e.g, 1 + 2 + 4 + 8 + ... = -1), which certainly cannot be done on the basis of convergence.
Well, what's the right way to understand infinite sums generally, or infinite decimals in particular, in the surreal numbers? In R, you can say e.g. that an infinite sum of nonnegative terms is the l.u.b. of the finite partial sums, but you can't do that in the surreals because you don't have lubs. You could handle infinite decimals specially, as per Scott's suggestion that .abcd... means {.a, .ab, .abc, .abcd, ... | .A, .aB, .abC, .abcD, ... } where I hope my notation is obvious, and indeed that will get you plausible answers for infinite decimals, but all that's doing is defining your notation so that infinite decimals denote (something like) the same real numbers as they usually do; it doesn't seem to shine much light on infinite series of surreal numbers in general.
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. 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?
Well, if h is 1/omega then maybe 1-h is the *simplest* thing satisfying some convenient property. As Scott says, that's what you get from his interpretation of infinite decimals in this case. (On the other hand, one can tweak Scott's definition a little -- e.g., add 2 rather than 1 to each final digit -- so that all infinite decimals, even terminating ones, "converge" to their conventional values, and that might be a better convention.)
I'm not sure how you'd make that work for infinite series generally, but I bet it would end up violating the first of those properties: you'd get cases where two series converge to a-1/omega and b-1/omega and their termwise sum converges to a+b-1/omega instead of a+b-2/omega.