Path: ...!weretis.net!feeder9.news.weretis.net!news.quux.org!eternal-september.org!feeder2.eternal-september.org!news.eternal-september.org!.POSTED!not-for-mail
From: WM <wolfgang.mueckenheim@tha.de>
Newsgroups: sci.math
Subject: Re: Incompleteness of Cantor's enumeration of the rational numbers
Date: Sun, 17 Nov 2024 18:38:00 +0100
Organization: A noiseless patient Spider
Lines: 35
Message-ID: <vhd9lq$obb0$1@dont-email.me>
References: <vg7cp8$9jka$1@dont-email.me>
 <1fca3a53-1cb4-4fd2-85b6-85e9b69ca23b@att.net> <vgtpmo$153hf$6@dont-email.me>
 <d17f7542-986e-4897-89b4-dccaf11d5311@att.net> <vh00jj$1m6co$1@dont-email.me>
 <97304048-24f5-4625-82a7-d17427f2f6e3@att.net> <vh0hta$1pmql$1@dont-email.me>
 <65febd06-662b-4fa4-9aa8-f7353a79a110@att.net> <vh2k9p$29cql$1@dont-email.me>
 <157a949d-6c19-4693-8cee-9e067268ae45@att.net> <vh35nd$2d81g$1@dont-email.me>
 <cb0c9917-09a9-45f0-8fe9-cd059fa82dde@att.net> <vh4itg$2o3vu$1@dont-email.me>
 <8165b44b-1ba5-429d-8317-0b043b214b53@att.net> <vh76bi$3bnde$1@dont-email.me>
 <d90452a5-965b-443f-9146-96cdf9b3906c@att.net> <vh8ft0$3jqb5$3@dont-email.me>
 <vh8g5e$3juug$1@dont-email.me> <vh8otd$3lhlt$2@dont-email.me>
 <vhb1is$6hbv$2@dont-email.me> <vhb1mu$6hbv$4@dont-email.me>
 <vhb32t$7ese$1@dont-email.me> <vhch7n$hge9$2@dont-email.me>
 <vhcieh$jjk5$1@dont-email.me> <vhcjg5$hdd4$3@dont-email.me>
 <vhckkh$k32g$1@dont-email.me> <vhcp0r$hge9$6@dont-email.me>
 <vhd7d7$nt37$1@dont-email.me>
MIME-Version: 1.0
Content-Type: text/plain; charset=UTF-8; format=flowed
Content-Transfer-Encoding: 8bit
Injection-Date: Sun, 17 Nov 2024 18:38:02 +0100 (CET)
Injection-Info: dont-email.me; posting-host="cec0708a7b2b5d7f33db81afdc17ed34";
	logging-data="798048"; mail-complaints-to="abuse@eternal-september.org";	posting-account="U2FsdGVkX18h+xvKyJ1Pe2wNPVc1mpsZnM8Yks4W+kc="
User-Agent: Mozilla Thunderbird
Cancel-Lock: sha1:fswd6pcpUcSxAUC75+L6UO9bgaI=
Content-Language: en-US
In-Reply-To: <vhd7d7$nt37$1@dont-email.me>
Bytes: 3224

On 17.11.2024 17:59, FromTheRafters wrote:
> WM pretended :
>> On 17.11.2024 12:38, FromTheRafters wrote:
>>> WM presented the following explanation :
>>>> On 17.11.2024 12:01, FromTheRafters wrote:
>>>>> WM was thinking very hard :
>>>>>> On 16.11.2024 22:33, Moebius wrote:
>>>>>>
>>>>>>> For example "aleph_0 - aleph_0" is not defined.
>>>>>>
>>>>>> Small wonder. ℵo means only infinitely many: |ℕ|, |ℚ|, and many 
>>>>>> others.
>>>>>> |ℕ|-|ℕ| however is defined.
>>>>>
>>>>> No, it is not.
>>>>
>>>> If sets are invariable then ℕ \ ℕ is empty.
>>>> If |ℕ| concerns only the elements of ℕ, then |ℕ|-|ℕ|= 0.
>>>
>>> So, you're saying that if I take aleph_zero natural numbers and I 
>>> remove the aleph_zero odd numbers from consideration in a new set, I 
>>> will have a new emptyset instead of E?
>>
>> Try to understand. "aleph_0 - aleph_0" is not defined.
> 
> Try to understand that |N| equals aleph_zero.

Of course. ℵo equals |ℕ|, equals |ℚ|, equals all countable sets. It is 
simply another name for infinitely "many". |ℕ| however is a fixed 
infinite number. Note that sets are invariable.

Regards, WM