Re: Replacement of Cardinality

On 8/30/2024 12:34 PM, Chris M. Thomasson wrote:

On 8/30/2024 3:13 AM, Moebius wrote:

Am 30.08.2024 um 09:21 schrieb joes:

Am Thu, 29 Aug 2024 13:37:03 +0000 schrieb WM:

Le 28/08/2024 à 17:06, joes a écrit :

Am Wed, 28 Aug 2024 13:04:57 +0000 schrieb WM:

Le 28/08/2024 à 06:20, joes a écrit :

Am Tue, 27 Aug 2024 19:57:39 +0000 schrieb WM:

>

The interval (0, x) cannot contain ℵo unit fractions unless x is
sufficiently large.

>

„Sufficient” meaning „not infinitesimal”.

>

No. "Sufficient" means that ℵo non-empty finite gaps between unit
fractions fit into the interval. That is much larger than
infinitesimal.

>

So… you agree.

>

Yes, I misread.

>

Thus: The interval (0, x) contains finitely many unit fractions only
for infinitesimal x.

>
Actually, it either contains NO unit fractions (if x = 0 or x is infinitesimal but > 0) or it contains infinitely many (ℵo) unit fractions.
>
Proof: If x = 0 then no unit fraction is in (0, x) (since in this case (0, x) is empty). If x is infinitesimal but > 0, then (by definition) for all n e IN: x < 1/n. Hence no unit fraction is in (0, x). If on the other hand x e IR, x > 0, and there is an unit fraction u such that u < x, then (0, x) contains (at least) the infinitely many unit fractions u, 1/(1/u + 1), 1/(1/u + 2), 1/(1/u + 3), ... (hence it contains ℵo unit fractions). qed
>
So no matter if x > 0 is infinitesimal or not: NUF(x) =/= 1.
>
Hence especially for all x e IR, x > 0: NUF(x) =/= 1. In fact, for all x e IR: NUF(x) e {0, aleph_0}.
>
WM is a crank.
>

 WM cranking a hurdy gurdy waiting for it to say the smallest unit fraction? lol. ;^D
 Hummm... I don't think WM is nearly as smart as the monkey grinder:
 https://www.thezorklibrary.com/history/monkey_grinder.html
 Oh shit. He is trolling us for sure. If we was as moronic as seems to

^^^^^^^^^^^^^^^^^^^^^^^^^                 ^^
Damn typo.
If _he_ was as moronic as seems to be
Well, perhaps the jokes on us wrt even giving WM the time of day.
ARGH!

be, well, he would need corks on his forks to prevent him from hurting himself and/or others. However, I have met new people commenting in these threads of doom. So, that's okay. :^)

[...]