Path: ...!weretis.net!feeder9.news.weretis.net!i2pn.org!i2pn2.org!.POSTED!not-for-mail
From: Richard Damon <richard@damon-family.org>
Newsgroups: sci.math
Subject: Re: How many different unit fractions are lessorequal than all unit
 fractions?
Date: Tue, 10 Sep 2024 21:37:12 -0400
Organization: i2pn2 (i2pn.org)
Message-ID: <9a0fb32f20ef76e6a71e2de2d35113f0efb676b3@i2pn2.org>
References: <vb4rde$22fb4$2@solani.org> <vb6o0r$3a4m1$2@dont-email.me>
 <7e1e3f62-1fba-4484-8e34-6ff8f1e54625@att.net> <vbabbm$24a94$1@solani.org>
 <06ee7920-eff2-4687-be98-67a89b301c93@att.net>
 <38ypmjbnu3EfnKYR4tSIu-WavbA@jntp>
 <34e11216-439f-4b11-bdff-1a252ac98f8f@att.net> <vbd56i$fqa0$1@dont-email.me>
 <vbdbq3$gdoe$2@dont-email.me> <vbes57$qdqo$2@dont-email.me>
 <27b3b5e088d82d4475c68a64f50a4bccac9c6f29@i2pn2.org>
 <vbesjo$27gfe$1@solani.org> <vbf0s9$qp1j$3@dont-email.me>
 <vbfpf5$utdu$2@dont-email.me> <vbh5qe$19a45$1@dont-email.me>
 <vbhdf6$1btm1$1@dont-email.me> <vbhdsh$1btm1$2@dont-email.me>
 <vbhj65$1bi3k$3@dont-email.me>
 <9eeba8a5041ce7ee48e5019d9e98d4ea38a1eb72@i2pn2.org>
 <vbhkmo$1bi3l$8@dont-email.me>
 <89ed6d8de6c20d65e869d384181b642309f63bc4@i2pn2.org>
 <vbkv1r$20uoi$4@dont-email.me>
 <7a991922c09e309450ac278f884091dfe716cae3@i2pn2.org>
 <vbmij2$2ce0a$4@dont-email.me>
 <20d29fc87d2bab70a7a7c18a3f79b093a4c2b647@i2pn2.org>
 <vbn322$2em19$1@dont-email.me>
MIME-Version: 1.0
Content-Type: text/plain; charset=UTF-8; format=flowed
Content-Transfer-Encoding: 8bit
Injection-Date: Wed, 11 Sep 2024 01:37:12 -0000 (UTC)
Injection-Info: i2pn2.org;
	logging-data="1605590"; mail-complaints-to="usenet@i2pn2.org";
	posting-account="diqKR1lalukngNWEqoq9/uFtbkm5U+w3w6FQ0yesrXg";
User-Agent: Mozilla Thunderbird
In-Reply-To: <vbn322$2em19$1@dont-email.me>
X-Spam-Checker-Version: SpamAssassin 4.0.0
Content-Language: en-US
Bytes: 4076
Lines: 54

On 9/9/24 11:08 AM, WM wrote:
> On 09.09.2024 13:49, joes wrote:
>> Am Mon, 09 Sep 2024 12:27:47 +0200 schrieb WM:
>>> On 08.09.2024 22:11, Richard Damon wrote:
>>>> On 9/8/24 3:48 PM, WM wrote:
>>>
>>>>> Select any gap between one of the first ℵo unit fractions and its
>>>>> neighbour. Call its size x. Then ℵo unit fractions cannot fit into the
>>>>> interval (0, x), independent of the actual size.
>>>> But that is changing the value of x in the middle of the problem which
>>>> isn't allowed.
>>>> Given that new x, we can choose a new set of Aleph_0 unit fractions
>>>> below that x.
>>> ℵo unit fractions are claimed to be smaller than every x > 0. If that is
>>> true then I can choose as the x one of the ℵo intervals between two of
>>> them.
>> More precisely: every positive x has infinitely many smaller unit
>> fractions (mind the quantifier order).
> 
> The quantifier order related to the problem is this: NUF(x) = ℵo means: 
> There exist ℵo unit fractions smaller than any x > 0. If this is not 
> true, then there are fewer. How many unit fractions are smaller than any 
> x > 0. THAT is the question. None. But all are differente. Hence there 
> must be a first one smaller than all other unit fractions. Note that 
> real points are in question. Real points fixed on the real line.

The number of unit fractions smaller than any positive x is Aleph_0. 
PERIOD. They are all diffferent.

That does NOT mean there must be a first, THAT is irrational finite 
thinking, which doesn't work on unbounded sets, and your using it has 
exploded your brain into smithereens by its contradictions.

> 
>  > A number is not an interval.
> 
> An interval has a length that can be expressed by a real number:
> 1/n - 1/(n+1) = x .
> 
> Then the interval (0, x) contains not all unit fractions, for instance 
> not 1/n.

But it contains almost all of them, an Aleph_0 of them while missing 
only a finite number of them.

A set of size Aleph_0 can (repeatedly) loose finite number of elements 
and maintain its size. This fact seems to be beyond your ability to 
understand.


> 
> Regards, WM
>>
>