Re: Incompleteness of Cantor's enumeration of the rational numbers (extra-ordinary)

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