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From: Richard Damon <richard@damon-family.org>
Newsgroups: comp.theory
Subject: Re: Cantor Diagonal Proof
Date: Tue, 8 Apr 2025 21:27:58 -0400
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On 4/8/25 6:59 PM, Andy Walker wrote:
> On 08/04/2025 21:19, Mr Flibble wrote:
>> I don't have to google anything: if you can multiply an infinitesimal by
>> 0.5 then it isn't an infinitesimal
> 
>      Are you under some strange impression that there can be only one
> infinitesimal?  If f is infinitesimal and g == 0.5*f, is g not also
> infinitesimal?  In the hyperreals and in the surreals, infinitesimals
> have their own algebra [and of course can be combined with (standard)
> reals in all the usual ways].
> 
>      FTAOD, a positive infinitesimal is a number f s.t. there is no
> integer N s.t. N*f > 1.  In the standard reals, there is no such number,
> but that is /only/ by fiat of the Archimedean axiom, and other number
> systems exist in which that axiom does not hold.
> 
>>                      -- it was twice as large as another
>> real and all real numbers can be multiplied by 0.5.
> 
>      Twice as large as another /number/.  Spot the difference.
> 

And then you get to the real weird fact, that when you get to 
real-scaled infintesimals (where you can talk about 0.5 * the base 
infinitesimal) you also get the fact that there is a strange gap between 
the "smallest" real-scaled-infinitisimal and zero, allowing us to create 
a second-level infinitesimal, and then a third, and so on.

Consistant definitions of Mathematics exist that handles all of that, 
but some things get a bit strange, and you need to take care of things 
that aren't an issue with "normal" Real Numbers.