Re: All infinities are countable in ordinary mathematics

On 05/04/2025 12:24 PM, Julio Di Egidio wrote:

On 04/05/2025 20:24, Ross Finlayson wrote:
>

The incompatibility of infinitesimals with the Archimedean principle
doesn't directly imply that all infinities must be countable in standard
mathematics.

>
I said in *ordinary* mathematics.  But you won't learn.
>
Julio
>

Oh, I didn't write that, in that dialog with one of those
mechanical reasoners "Google Gemini", I only wrote the
paragraphs starting "Thanks GG.".
That there are multiple models of continuous domains:
has that "least upper bound" isn't necessarily "complete
orderd field", instead "iota-values" or "standard infinitesimals".
If you're going to consider mathematical infinitesimals, they might
as well be the "standard infinitesimals" we were taught to ignore
in pre-calculus, like Newton's fluxions or Leibnitz' raw differentials,
as with regards to the non-nilpotent infinitesimals and non-nilsquare.
Most people's ideas of "infinitesimals" is only exactly
"constant monotone strictly increasing" from lower to upper bound,
"constant, monotone, strictly increasing, vanishing" values.
Then sometimes there's talk about "real-valued" instead of
"the complete ordered field" to talk about continuous domains,
that have a usual common formalism in the complete ordered field,
where it's axiomatized that there's least-upper-bound property
and it's axiomatizezd that measure 1.0 is a thing, that can be
justified for itself from line-real/iota-values getting those
for free.
These are objects of mathematics, they exist regardless being
defined away, naive positivist.