Path: ...!npeer.as286.net!npeer-ng0.as286.net!weretis.net!feeder8.news.weretis.net!eternal-september.org!feeder3.eternal-september.org!news.eternal-september.org!.POSTED!not-for-mail
From: "Chris M. Thomasson" <chris.m.thomasson.1@gmail.com>
Newsgroups: sci.logic,sci.math
Subject: Re: Replacement of Cardinality
Date: Sat, 3 Aug 2024 14:51:35 -0700
Organization: A noiseless patient Spider
Lines: 47
Message-ID: <v8m8p8$3l66h$1@dont-email.me>
References: <hsRF8g6ZiIZRPFaWbZaL2jR1IiU@jntp>
 <P04FAosjolyjDfgV0JPXWx1mF6o@jntp>
 <40165884-df8f-4614-8644-9161d72fd1cb@att.net>
 <-qUZ96ARwcjh9QPfyWRnijjNwoY@jntp>
 <6b837540-3d9a-4b8e-9a70-88d52e81a1a4@att.net>
 <xQQT0K_Q_k2FbMcCUXF8j3CEg84@jntp>
 <9822f5da-d61e-44ba-9d70-2850da971b42@att.net>
 <p36L63dXamDAkHDhkZhDKqx-h-o@jntp>
 <d8bbe664-a601-4590-9a7f-d5312b4dae54@att.net> <v8j55c$2u09m$2@dont-email.me>
 <w2KNEc6WpgYmvVtoH_VZTRkLnUg@jntp>
MIME-Version: 1.0
Content-Type: text/plain; charset=UTF-8; format=flowed
Content-Transfer-Encoding: 8bit
Injection-Date: Sat, 03 Aug 2024 23:51:36 +0200 (CEST)
Injection-Info: dont-email.me; posting-host="c5d5c7fe7b787c73b28f56e08a59c030";
	logging-data="3840209"; mail-complaints-to="abuse@eternal-september.org";	posting-account="U2FsdGVkX1/Vixl2FQB/2A4B4icSitWmUy+4sB8INvE="
User-Agent: Mozilla Thunderbird
Cancel-Lock: sha1:4VAU/Lbt1wpXvZ0z/IwjWFe+Q3k=
In-Reply-To: <w2KNEc6WpgYmvVtoH_VZTRkLnUg@jntp>
Content-Language: en-US
Bytes: 2784

On 8/3/2024 7:25 AM, WM wrote:
> Le 02/08/2024 à 19:31, Moebius a écrit :
>> For each and every of these points [here referred to with the variable 
>> "x"]: NUF(x) = ℵ₀ .
> 
> I recognized lately that you use the wrong definition of NUF.
> Here is the correct definition:
> There exist NUF(x) unit fractions u, such that for all y >= x: u < y.
> Note that the order is ∃ u ∀ y.
> NUF(x) = ℵ₀ for all x > 0 is wrong. NUF(x) = 1 for all x > 0 already is 
> wrong since there is no unit fraction smaller than all unit fractions.
> ℵ₀ unit fractions need ℵ₀*2ℵ₀ points above zero.

0->(...)->(1/1)

Contains infinite unit fractions.

0->(...)->(1/2)->(1/1)

Contains infinite unit fractions.

0->(...)->(1/3)->(1/2)->(1/1)

Contains infinite unit fractions.

However, (1/3)->(1/1) is finite and only has three unit fractions 
expanded to:

(1/3)->(1/2)->(1/1)

Just like the following has four of them:

(1/4)->(1/3)->(1/2)->(1/1)


(0/1) is not a unit fraction. There is no smallest unit fraction. 
However, the is a largest one at 1/1.

A interesting part that breaks the ordering is say well:

(1/4)->(1/2)

has two unit fractions. Then we can make it more fine grain:

(1/4)->(1/2) = ((1/8)+(1/8))->(1/4+1/4)

;^)