Path: ...!eternal-september.org!feeder3.eternal-september.org!news.eternal-september.org!.POSTED!not-for-mail
From: WM <wolfgang.mueckenheim@tha.de>
Newsgroups: sci.logic
Subject: Re: Incompleteness of Cantor's enumeration of the rational numbers
 (extra-ordinary)
Date: Sat, 14 Dec 2024 10:50:52 +0100
Organization: A noiseless patient Spider
Lines: 37
Message-ID: <vjjkds$3ukck$1@dont-email.me>
References: <vg7cp8$9jka$1@dont-email.me> <vhn3po$jvo1$1@dont-email.me>
 <vhn420$jf6v$3@dont-email.me> <vhpg51$13soc$1@dont-email.me>
 <vhpnrb$15239$1@dont-email.me> <vhs2gn$1kjtc$1@dont-email.me>
 <vhs4ue$1ku9t$1@dont-email.me> <vhv6or$280s6$1@dont-email.me>
 <vhvbjb$28n6o$1@dont-email.me> <vi1dbj$2moon$1@dont-email.me>
 <vi224l$2pgrd$1@dont-email.me> <vi4383$3csd4$2@dont-email.me>
 <vi4a6c$3dt4s$2@dont-email.me> <vi6p1l$3uoti$1@dont-email.me>
 <vi6unr$3v0dn$5@dont-email.me> <vihd3l$2d9fk$1@dont-email.me>
 <vihfai$2cnof$1@dont-email.me> <vijrru$37ce1$1@dont-email.me>
 <vikh9k$3cua3$1@dont-email.me> <viml28$6j3$1@dont-email.me>
 <0b1bb1a1-40e3-464f-9e3d-a5ac22dfdc6f@tha.de>
 <95183b4d9c2e32651963bac79965313ad2bfe7e8@i2pn2.org>
 <vj6vhh$elqh$2@dont-email.me>
 <33512b63716ac263c16b7d64cd1d77578c8aea9d@i2pn2.org>
 <vj9s4i$11a3p$1@dont-email.me> <vjam6d$1700v$1@dont-email.me>
 <vjc65g$1i9vk$3@dont-email.me> <vjf7kl$2s7e5$1@dont-email.me>
 <vjfmq3$2upa9$3@dont-email.me> <vjjh0o$3u5ec$1@dont-email.me>
MIME-Version: 1.0
Content-Type: text/plain; charset=UTF-8; format=flowed
Content-Transfer-Encoding: 8bit
Injection-Date: Sat, 14 Dec 2024 10:50:52 +0100 (CET)
Injection-Info: dont-email.me; posting-host="b67ed49269f1575d0aac6d4d8c698b63";
	logging-data="4149652"; mail-complaints-to="abuse@eternal-september.org";	posting-account="U2FsdGVkX18xuPkhaBUhE44GNYM4WrS3nBT6+HMW2Zk="
User-Agent: Mozilla Thunderbird
Cancel-Lock: sha1:6nC64Skx3E6guRwghWPvmnzGUbU=
Content-Language: en-US
In-Reply-To: <vjjh0o$3u5ec$1@dont-email.me>
Bytes: 3425

On 14.12.2024 09:52, Mikko wrote:
> On 2024-12-12 22:06:58 +0000, WM said:

>>>> In mathematics, a set A is Dedekind-infinite (named after the German 
>>>> mathematician Richard Dedekind) if some proper subset B of A is 
>>>> equinumerous to A. [Wikipedia].
>>>
>>> Do you happen to know any set that is Dedekind-infinite?
>>>
>> No, there is no such set.
> 
> The set of natural numbers, if there is any such set,

If ℕ is a set, i.e. if it is complete such that all numbers can be used 
for indexing sequences or in other mappings, then it can also be 
exhausted such that no element remains. Then the sequence of 
intersections of endsegments
E(1), E(1)∩E(2), E(1)∩E(2)∩E(3), ...
loses all content. Then, by the law
∀k ∈ ℕ : ∩{E(1), E(2), ..., E(k+1)} = ∩{E(1), E(2), ..., E(k)} \ {k}
the content must become finite.

> is Dedekind-infinte:
> the successor function is a bijection between the set of all natural
> numbers and non-zero natural numbers.

This "bijection" appears possible but it is not. This is better 
demonstrated by the "bijection" between the sets ℕ = {1, 2, 3, ...} and 
D = {10n | n ∈ ℕ}. It is contradicted because for every interval (0, n] 
the relative covering is not more than 1/10, and there are no further 
numbers 10n beyond all natural numbers n. The sequence 1/10, 1/10, 1/10, 
.... has limit 1/10.

Regards, WM

>