Path: ...!news.mixmin.net!proxad.net!feeder1-2.proxad.net!usenet-fr.net!news.gegeweb.eu!gegeweb.org!pasdenom.info!from-devjntp
Message-ID: <6WIT-GYNvuMQ6ADdNvBdVKBkQ1c@jntp>
JNTP-Route: news2.nemoweb.net
JNTP-DataType: Article
Subject: Re: Replacement of Cardinality
References: <hsRF8g6ZiIZRPFaWbZaL2jR1IiU@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> <F6pqEi9Vg1YMcYTcIPQNs6NU_vI@jntp>
 <4f606ef2-ef6c-487b-b959-d109e374929f@att.net> <vpb42BOZYYy79eBYzCXpUbsjGQc@jntp>
 <f5086d19-ab91-429a-9dfe-2325e56c97a4@att.net>
Newsgroups: sci.logic,sci.math
JNTP-HashClient: uO04DEw7HcEfbv2KOFgnD472h10
JNTP-ThreadID: KFm3f7lT2HjaTSiMfnv5xqZoSBw
JNTP-Uri: http://news2.nemoweb.net/?DataID=6WIT-GYNvuMQ6ADdNvBdVKBkQ1c@jntp
User-Agent: Nemo/0.999a
JNTP-OriginServer: news2.nemoweb.net
Date: Sun, 04 Aug 24 18:13:22 +0000
Organization: Nemoweb
JNTP-Browser: Mozilla/5.0 (Windows NT 10.0; Win64; x64) AppleWebKit/537.36 (KHTML, like Gecko) Chrome/127.0.0.0 Safari/537.36
Injection-Info: news2.nemoweb.net; posting-host="82b75c1d0a83e677ff646b52485f72f8b23749df"; logging-data="2024-08-04T18:13:22Z/8975610"; posting-account="217@news2.nemoweb.net"; mail-complaints-to="julien.arlandis@gmail.com"
JNTP-ProtocolVersion: 0.21.1
JNTP-Server: PhpNemoServer/0.94.5
MIME-Version: 1.0
Content-Type: text/plain; charset=UTF-8; format=flowed
Content-Transfer-Encoding: 8bit
X-JNTP-JsonNewsGateway: 0.96
From: WM <wolfgang.mueckenheim@tha.de>
Bytes: 3862
Lines: 77

Le 04/08/2024 à 18:39, Jim Burns a écrit :
> On 8/4/2024 11:29 AM, WM wrote:
>> Le 03/08/2024 à 21:54, Jim Burns a écrit :
>>> On 8/3/2024 10:23 AM, WM wrote:
> 
>>>> 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.
>>
>>> Here is an equivalent definition:
>>> There exist NUF(x) unit fractions u, such that
>>> u < x
>>>
>>>> Note that the order is ∃ u ∀ y.
>>>
>>> The order is ∀x ∃u ∀y

When all x are involved, the universal quantifier is usually not written.
> 
> The order of the claim which you (WM) address
> in an attempt to "prove" dark numbers is
> ∀ᴿx > 0:
> ∃U ⊆ ⅟ℕ ∧ |U| = ℵ₀:
> ∀ᴿy ≥ x:
> y >ᵉᵃᶜʰ U
> 
> That claim and the following claim are
> either both true or both false.
> ∀ᴿx > 0:
> ∃U ⊆ ⅟ℕ ∧ |U| = ℵ₀:
> x >ᵉᵃᶜʰ U

More of interest are these two claims which are not both true or both 
false:
For every x there is u < x.
There is u < x for every x.
The latter is close to my function:
There are NUF(x) u < x.

> Your recently corrected definition of NUF is
> NUF(x) =
> |{u ∈ ⅟ℕ: ∀ᴿy ≥ x: y > u}|
> 
> That definition is equivalent to
> NUF(x) =
> |{u ∈ ⅟ℕ: x > u}|
> 
> Note that,
> for x > 0, {u ∈ ⅟ℕ: x > u}
> is maximummed and down.stepped and non.max.up.stepped.
> For x > 0:  |{u ∈ ⅟ℕ: x > u}| = ℵ₀
> 
> The claim you (WM) use
> ∃U ⊆ ⅟ℕ ∧ |U| = ℵ₀:
> ∀ᴿx > 0:
> ∀ᴿy ≥ x:
> y >ᵉᵃᶜʰ U
> 
> is an unreliable quantifier shift from
> the claim we make

What claim you make is not of interest to me. I express that no u can be 
smaller than all x but that some u can be smaller than many x.
You express that for all x, there is a smaller u. Both are very different.

> Only you (WM) think that ∃u ∀x>0: u < x
> follows from ∀x>0 ∃u: u < x,

Not at all! Please spare these insults! Your claim concerns only definable 
x. For ℵo*2^ℵo undefinable points x it is wrong. My claim concerns all 
x. 

Regards, WM