Re: Cantor Diagonal Proof

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