Re: Cantor Diagonal Proof

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.