Re: Infintesimals

From: Shmuel (Seymour J.) Metz (spamtrap_at_library.lspace.org.invalid)
Date: 07/01/04 

Date: Thu, 01 Jul 2004 18:24:33 -0300

In <40e1f568_1@newsfeed.slurp.net>, on 06/29/2004
   at 04:03 PM, "Tralfaz" <tralfaz@emailblackhole.com> said:

>While writing a paper in my philosophy class on Zeno's Paradox, I
>came across a few moder papers "proving" the idea of infintesimals

You can't: you can only define them.

>numbers that are not zero yet are smaller than any positive real

ITYM smaller in absolute value than any standard real.

>Admittedly, if these do in fact exist it would resolve many of the
>issues some people have with infinite series (e.g. 0.999... =1)

No, it doesn't resolve any of them. The issues that they have are
psychological rather than Mathematical.

>Many of the papers I see "prove" these numbers exist by noting that
>if an infinite series converges (like sigma 9x10^(-n))then for any
>finite partial sum, the sum is less than the limit of the sums (1).
>Therefore the sum never "gets to" the limit.

That isn't even wrong.

>This gap is then "filled in"

There is no gap. There is a sequence of partial sums and a limit.

>by infintesimals that have positive magnitude but are not real

What is that supposed to mean? It's just arm waving.

>Given this assumption,

What assumption?

>the fact that 0.999...=1 "proves" infintesimals exist

No.

>My question is: what is the opinion of other mathematicians as to
>the existance of these types of numbers

What types of numbers? Something involving slinging words around
without understanding, or infinitesimals as defined by Mathematicians?
Robinson defined a rigorous concept of infinitesimals decades ago, but
it has nothing to do with mystical gaps. 

-- 
     Shmuel (Seymour J.) Metz, SysProg and JOAT
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