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

On 1/9/25 11:59 AM, WM wrote:

On 09.01.2025 17:11, FromTheRafters wrote:

WM wrote on 1/9/2025 :

On 09.01.2025 13:27, FromTheRafters wrote:

WM wrote :

On 09.01.2025 01:07, joes wrote:

Am Wed, 08 Jan 2025 22:57:52 +0100 schrieb WM:

>

The rule is for n there is n+1. But the successor is not created but
does exist. How far do successors reach? Why do they not reach to ω-1?
Where do they cease before?

They don't cease. They simply aren't in the same league, if you will.

>
Cantor will. Every set of numbers of the first and second number class has a smallest element. Hence they all are on the ordinal line.

>
Zero is the smallest in the natural number class, omega is the smallest of the infinite number class. Neither has a predecessor in its class.

>
Are the natural numbers fixed or do they evolve?

>
Neither

 There is no third alternative.

No, because you alternatives are just a type error of your logic. "Nubmers" or "Sets" don't evolve. You seem to THINK that sets, particularly "potentially infinite" set "evolve" in that numbers get added to them as you move along the generator, but the set doesn't change, only our knowledge of the set.

 

they are the smallest infinite set.

 The set of prime numbers is infinite but smaller because it is a proper subset. It has less than 1 % content.

It may SEEM smaller, but it turns out it is of the same countable infinite cardinality.

 Regards, WM