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From: Julio Di Egidio <julio@diegidio.name>
Newsgroups: sci.math
Subject: Re: All infinities are countable in ordinary mathematics
Date: Sun, 4 May 2025 18:54:35 +0200
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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