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From: "Chris M. Thomasson" <chris.m.thomasson.1@gmail.com>
Newsgroups: sci.math
Subject: Re: Incompleteness of Cantor's enumeration of the rational numbers
 (extra-ordinary)
Date: Mon, 13 Jan 2025 12:55:19 -0800
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On 1/13/2025 9:17 AM, WM wrote:
> 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.

No. double all of the natural numbers:

{ 1 * 2, 2 * 2, 3 * 2, 4 * 2, ... } = { 2, 4, 6, 8, ... }

All of those results are natural numbers. They were already there...


>>
>> 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
>