Re: Replacement of Cardinality

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 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. :^)
WM not as smart as, or is he? lol.
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MONKEY GRINDER
Avoid this nightmare at any cost! Spawn of a carnival necromancer, the monkey grinder (also known as an Organ Grinder) can blast minds to jelly with its powerful Sense Organ. A single blast is enough to leave its victim a gibbering wreck with a shattered mind. These creatures seem to have powerful control over energy and space, allowing them to come and go as they please. Their arrival is heralded by a burst of hollow laughter and a faint, electrical tension in the air, followed by planes of energy; their patterns of intersection whirling around a vortex of laughter, growing in power and malevolence as the entire form of the monkey grinder suddenly appears. If one is strong enough to defeat it in combat, its body dissolves in a puff of steam, leaving its hurdy-gurdy behind.
Extremely arrogant, monkey grinders will not hesitate to destroy a lowly peasant with the help of its Sense Organ. Deceptively eloquent in both manner and speech, these loathsome creatures actually possess little intelligence, and suffer an illiteracy rate of 103% (which explains why many have to dictate their threatening messages coupled with crazed laughter). This stupidity is, in fact, their Achilles' Heel. By tricking the creature into thinking it is receiving a gift (they love surprises), one could actually give it a one-way portal to the Planes of TransInfinite Splendor, banishing it forever.
Monkey grinders especially despise nymphs, quickly snatching any they see out of the air and crushing the "miserable pests" in their fists.
During the Great Monster Uprising of the Second Age of Magic, monkey grinders were reported in the following regions: The Gray Mountains Asylum, Pheebor, Fields of Frotzen, Forest of Youth, Gurth Woodland Trial, Crab Island, The Glass Maze, Bozbarland, The Old Lingolf House, Largoneth Castle, and the G.U.E. Tech Training Grounds in the Ethereal Plane of Atrii.
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Also, well, WM on a good day?
https://youtu.be/SKDX-qJaJ08
Humm...