RE: If 0.999 (etc.) = 1, does 1 - 0.999 = 0?
March 4, 2012 at 2:56 pm
(This post was last modified: March 4, 2012 at 3:10 pm by Categories+Sheaves.)
There are a lot of practical (and aesthetic) reasons to not permit the existence of infinitesimals. Which is why you don't see infinitesimals given much attention.
The most popular modern formulation of analysis with infinitesimals (Abraham Robinson's Hyperreal numbers) involves the construction of an ultrafilter on the natural numbers, and the uniqueness of the Hyperreals as a system of numbers is equivalent to the continuum hypothesis. It's all well-defined and whatnot, but it's a bit further... out there. In order to divide by infinitesimals, you need to have infinitely large numbers. Most statements about the hyperreal numbers are true if and only if a 'standard' version of the statement (about the real numbers) is also true, so even if infinitesimal numbers don't exist, they do still offer an improved methodology for proving things about 'regular' numbers (a lot of difficult results in standard analysis have one- or two-line proofs in the hyperreals).
So I mean, you can talk about infinitesimals and still make sense. But they're a rather unpopular concept, and the machinery you'd need to talk about them is sort of advanced.
The most popular modern formulation of analysis with infinitesimals (Abraham Robinson's Hyperreal numbers) involves the construction of an ultrafilter on the natural numbers, and the uniqueness of the Hyperreals as a system of numbers is equivalent to the continuum hypothesis. It's all well-defined and whatnot, but it's a bit further... out there. In order to divide by infinitesimals, you need to have infinitely large numbers. Most statements about the hyperreal numbers are true if and only if a 'standard' version of the statement (about the real numbers) is also true, so even if infinitesimal numbers don't exist, they do still offer an improved methodology for proving things about 'regular' numbers (a lot of difficult results in standard analysis have one- or two-line proofs in the hyperreals).
So I mean, you can talk about infinitesimals and still make sense. But they're a rather unpopular concept, and the machinery you'd need to talk about them is sort of advanced.
