Path: ...!news.mixmin.net!weretis.net!feeder8.news.weretis.net!pasdenom.info!from-devjntp
Message-ID: <vg44QVKbPSR4U0Tq71L-fg5yqgM@jntp>
JNTP-Route: news2.nemoweb.net
JNTP-DataType: Article
Subject: Re: Replacement of Cardinality
References: <hsRF8g6ZiIZRPFaWbZaL2jR1IiU@jntp> <881fc1a1-2e55-4f13-8beb-94d1f941b5af@att.net>
Newsgroups: sci.logic,sci.math
JNTP-HashClient: ABo5Ooj4UZmp2v-D_36z4BMps3I
JNTP-ThreadID: KFm3f7lT2HjaTSiMfnv5xqZoSBw
JNTP-Uri: http://news2.nemoweb.net/?DataID=vg44QVKbPSR4U0Tq71L-fg5yqgM@jntp
User-Agent: Nemo/0.999a
JNTP-OriginServer: news2.nemoweb.net
Date: Sun, 28 Jul 24 12:17:51 +0000
Organization: Nemoweb
JNTP-Browser: Mozilla/5.0 (Windows NT 10.0; Win64; x64) AppleWebKit/537.36 (KHTML, like Gecko) Chrome/126.0.0.0 Safari/537.36
Injection-Info: news2.nemoweb.net; posting-host="82b75c1d0a83e677ff646b52485f72f8b23749df"; logging-data="2024-07-28T12:17:51Z/8966954"; 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: 2802
Lines: 57

Le 27/07/2024 à 19:34, Jim Burns a écrit :

> If ℕ has fewer elements than ℕ∪{ℕ}
> then
> |ℕ| ∈ ℕ

|ℕ| = ω-1 ∈ ℕ

> ℕ has fewer elements than ℕ

ℕ has ω-1 elements.
> 
> Because ℕ does not have fewer elements than ℕ
> ℕ does not have fewer elements than ℕ∪{ℕ}
> and the rule of subsets is broken.

ℕ = {1, 2, 3, ..., ω-1} = {1, 2, 3, ..., |ℕ|}
> 
>> _The rule of subset_ proves that
> 
> To make a claim
>   is not sufficient
> to make a proof.
> 
> To make a finite sequence of claims
>   such that no claim is first.false
>   is sufficient
> to make a proof.

First false is your claim that |ℕ| is larger than all elements of ℕ. 
ℕ counts its elements.
> 
> The most that is true here is that
> the rule of subset _claims_ without proof that
> 
>> every proper subset has
>> less elements than its superset.

The proof is easy. Since the superset has at least one more element than 
its proper subset, it has more elements than its proper subset.

> 
> ⎛ In English, grammatically speaking,
> ⎜ it is never correct to say "less <plural.noun>"
> ⎜
> ⎜ English has mass nouns (Stoffnamen)
> ⎜ "less rock" ...
> ⎜ and count nouns (zählbare Substantive)
> ⎜ "one rock", "fewer rocks" ...
> ⎜ Only count nouns have a plural.
> ⎜ Only mass nouns are modified by "less".
> ⎝ "Less rocks" and "lescs elements" are never correct.

Thank you, I will try to remember it.

Regards, WM