On Sat, Nov 17, 2012 at 6:25 AM, Gareth McCaughan <gareth.mccaughan@pobox.com> wrote:
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.
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.
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. Andy