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

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.

Our sets do not change.
Our set ℕ does not change.
 

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?

 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.

 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.

 Which elements are in ℕ doesn't change.

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