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From: WM <wolfgang.mueckenheim@tha.de>
Newsgroups: sci.math
Subject: Re: Incompleteness of Cantor's enumeration of the rational numbers
 (extra-ordinary)
Date: Mon, 13 Jan 2025 18:17:49 +0100
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On 13.01.2025 17:33, Jim Burns wrote:
> On 1/12/2025 2:39 PM, WM wrote:
>> On 12.01.2025 20:33, Jim Burns wrote:
>>> On 1/12/2025 10:54 AM, WM wrote:
> 
>>>> No, it depends on completeness.
>>>
>>> It is completely true
>>> that each natural number is a natural number   and
>>> that only natural numbers are natural numbers.
>>
>> and that nothing fits between them and ω.
> 
> Yes.

Therefore doubling of all natural numbers creates numbers larger than ω.
> 
> ω is first infiniteᵒʳᵈ.
> No infiniteᵒʳᵈ is before ω
> No finiteᵒʳᵈ is after ω

Right.
> Regular distances in ⦅0,ω⦆ multiplied by 2
> remain regular distances in ⦅0,ω⦆, not.in ⟦ω,2ω⦆

Doubling of all n deletes the odd numbers but cannot change the number 
of numbers, therefore creates even numbers. They do not fit below ω. 
Remember: nothing fits between them and ω.

Regards, WM