RE: [split] 0.999... equals 1
February 25, 2012 at 2:28 am
(This post was last modified: February 25, 2012 at 2:29 am by Categories+Sheaves.)
(February 2, 2012 at 9:58 am)CliveStaples Wrote: (But I suspect that even in the hyperreals, .999... = 1.)
Mmk. Goldblatt's Lectures on the Hyperreals came in the mail this week...
Without getting into any business with ultrafilters, it turns out that there are too many elements in *R that I would have wanted .999... to be.
The issue is that we don't know whether .999... is referring to the equivalence class of [.9, .99, .999, ...] or [.99, .9999, .999999, ...] etc. (yes, these are two distinct elements of *R, and the first is less than the second).
So if '.999...' is a well-defined symbol for a single thing, we'd have to treat it as an element of R and not *R - R...
So: Touché. If our writing '.999...' is to stand any chance of being intelligible, .999... = 1