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

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

On 08.01.2025 21:07, Jim Burns wrote:

On 1/8/2025 9:35 AM, WM wrote:

On 08.01.2025 12:04, FromTheRafters wrote:

WM formulated on Wednesday :

 

If ω exists, then ω-1 exists.

Wrong.

A set like ℕ has a fixed number of elements.

Yes.

|ℕ| is invariable. |ℕ| = |ℕ|/2 is wrong.

No, |N| is not a member of itself and does not behave like it.

If ω-1 does not exist,
what is the fixed border of existence?

 
Membership in ℕ is determined by ℕ.rule.compliance,
not by position relative to a border.element.
Each object complying with the ℕ.rule is in ℕ Each object not.complying
is not.in ℕ

 
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.

Compliance and non-compliance do not change. Membership does not
change.

Then tell me if n = 7 exists and n = ω-1 does not exist where the border
lies.

Exactly omega is the "border", but this suggests a wrong mental image,
as it has to you.

No element is the border.element because each element is smaller.than
another, fuller element, and so, not on the border.

"And so on" can only happen, when the elements are created. Potential
infinity.

No, it is a finite expression of an infinity.

Which elements are in ℕ doesn't change.

--
Am Sat, 20 Jul 2024 12:35:31 +0000 schrieb WM in sci.math:
It is not guaranteed that n+1 exists for every n.