Path: ...!eternal-september.org!feeder3.eternal-september.org!news.eternal-september.org!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
 (extra-ordinary)
Date: Tue, 14 Jan 2025 09:58:22 +0100
Organization: A noiseless patient Spider
Lines: 48
Message-ID: <vm58vd$2b5ia$4@dont-email.me>
References: <vg7cp8$9jka$1@dont-email.me> <vllfof$2n0uj$2@dont-email.me>
 <07258ab9-eee1-4aae-902a-ba39247d5942@att.net> <vlmst2$2vjr0$3@dont-email.me>
 <1ebbc233d6bab7878b69cae3eda48c7bbfd07f88@i2pn2.org>
 <vlo5f4$39hil$2@dont-email.me>
 <4c89380adaad983f24d5d6a75842aaabbd1adced@i2pn2.org>
 <vloule$3eqsr$1@dont-email.me>
 <ffffed23878945243684de7f2aa9aaaf29564508@i2pn2.org>
 <vlrej9$2m5k$1@dont-email.me> <d6ed4797-65e8-4004-853c-f07a37af0c11@att.net>
 <vls4j6$7v2k$3@dont-email.me>
 <adef9ec5c327614374fdc3c3cc55d7a753e28a36@i2pn2.org>
 <vltfo8$heoh$5@dont-email.me>
 <7fc40cc2dbd42016a62aa0374d545e9e787a7da3@i2pn2.org>
 <vltu35$lav6$1@dont-email.me> <vlv88i$sa4k$2@dont-email.me>
 <vm0dl6$15ccj$2@dont-email.me>
 <b27df55b2cd58e362834a1aa0756e860cb5f31c9@i2pn2.org>
 <vm1vsc$1kllp$1@dont-email.me> <vm264g$1l95f$1@dont-email.me>
 <vm322j$1pmu9$1@dont-email.me> <e8fcb812-242f-4563-bdeb-ea3221b0aaf2@att.net>
 <vm3iib$1s9lc$2@dont-email.me> <309b1b88-6eca-4e27-b1d6-cd927203286f@att.net>
MIME-Version: 1.0
Content-Type: text/plain; charset=UTF-8; format=flowed
Content-Transfer-Encoding: 8bit
Injection-Date: Tue, 14 Jan 2025 09:58:22 +0100 (CET)
Injection-Info: dont-email.me; posting-host="808a3c7120e379c601c055d8bd308c06";
	logging-data="2463306"; mail-complaints-to="abuse@eternal-september.org";	posting-account="U2FsdGVkX19stZ9duIAKJ8U3ZGqmTybkwvP1ZrUVYYQ="
User-Agent: Mozilla Thunderbird
Cancel-Lock: sha1:IyItvhN9MXM0XHrMkLEfAvcxAGY=
Content-Language: en-US
In-Reply-To: <309b1b88-6eca-4e27-b1d6-cd927203286f@att.net>
Bytes: 3065

On 13.01.2025 21:44, Jim Burns wrote:
> On 1/13/2025 12:29 PM, WM wrote:
>> On 13.01.2025 18:06, Jim Burns wrote:
>>> On 1/13/2025 7:48 AM, WM wrote:

> ℕ is only invariable in the sense which we use.

That is potential infinity.

> However,
> you (WM) are convinced that
> a set (such as ℕ) larger than
> any set with sets.different.in.size.by.one
> changes (has elements inserted or deleted)
> in order to not.change.in.size.by.one.

In actual infinity all elements are invariable.
>
> 
>>> ⎜ #⟦0,𝔑⦆ = #ℕ
>>> ⎜
>>> ⎜ ⟦0,𝔑+1⦆ is 'fixed', too.
>>
>> ω
> 
> Is ω = ⟦0,𝔑+1⦆ your 'fixed' (our 'finite')?

ω is the first infinite ordinal by definition.

> What about ω+1 and ω+2?

Infinite ordinals too.
> 
>>> ⎜ ⟦0,𝔑+1⦆ ⊆ ℕ
>>
>> No.
> 
> Yes.

No.

> ∀𝔑 ∈ ℕ: #⟦0,𝔑⦆ < #⟦0,𝔑+1⦆ ≤ #ℕ

No. The sequence of endsegments can get empty by
∀k ∈ ℕ : E(k+1) = E(k) \ {k+1} . There is a sharp end.

Regards, WM