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From: Andy Walker <anw@cuboid.co.uk>
Newsgroups: comp.theory
Subject: =?UTF-8?Q?Re=3A_Definition_of_real_number_=E2=84=9D_--infinitesimal?=
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Date: Fri, 29 Mar 2024 13:23:28 +0000
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On 29/03/2024 04:29, olcott wrote:
> x is said to be infinitesimal
> if, and only if, |x| < 1/n for all integers n.
> https://en.wikipedia.org/wiki/Hyperreal_number

	That's for the hyperreals;  there's a clue in the URL.
There are no such "x" in R, by the Archimedean axiom.

> 0.999... specifies infinitesimally < 1.0

	No it doesn't.  It specifies different things in different
number systems, which is why mathematicians don't use that notation
in contexts where there could be ambiguity.

> and math guys have no way to say that so they
> simply round up to 1.0

	You've just referred to some "math guys" -- the proponents of
the hyperreals -- who say exactly what they mean by "infinitesimal".
You could equally have referred to the surreals [qv] where similar
statements are made and explained by "math guys".  Maths has moved
on over the past few centuries.  You and Wij need to move on with
the "math guys".

-- 
Andy Walker, Nottingham.
    Andy's music pages: www.cuboid.me.uk/andy/Music
    Composer of the day: www.cuboid.me.uk/andy/Music/Composers/Gottschalk