Re: All infinities are countable in ordinary mathematics

On 04/05/2025 17:17, Ross Finlayson wrote:

On 05/04/2025 06:27 AM, Julio Di Egidio wrote:

If there exist definite transfinite numbers, then their
reciprocals must be infinitesimals, not zero.  Which is
good, as infinitesimals necessitate specific additional
laws, reciprocally making the transfinite sharper.  And
one issue is immediately apparent: infinitesimals are
not compatible with the Archimedean principle.  Ergo,
all infinities are countable in ordinary mathematics.

 Pythagorean Archimedean [bollocks]
 A usual account of infinity has that it's not ordinary,
rather, per Mirimanoff, extra-ordinary, then that it's
fragments or extensions, the model of integers.

Are you aware of the fact that the least upper-bound
property, which is an axiom of the standard theory of
real numbers, and a formalisation of the notion of
continuum with it, *implies the Archimedean property*?
Indeed, there is ordinary and there is extra-ordinary,
and *invalidly* then *inconsistently* mixing results
is the problem there.
(But already the prefix "extra", which is necessarily
extra-to given something, here the "ordinary", should
at least make *you* pause, and rather warn you that you
have it upside-down, what "ordinary mathematics" even
is, as per the usual inversion of all that counts.
Conversely, your balderdash, here as elsewhere, always
eventually back to your blind take-everything and fully
prosaic Platonism, remains the other side of the very
same mangled/fraudulent coin.  Strictly speaking.)
Julio