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

Am Tue, 24 Dec 2024 11:42:56 +0100 schrieb WM:

On 23.12.2024 15:31, Richard Damon wrote:

On 12/23/24 3:52 AM, WM wrote:

On 23.12.2024 01:05, FromTheRafters wrote:

WM brought next idea :
>

The function E(n) decreases from infinity to zero because in set
theory ℕ \ {1, 2, 3, ...} =  { } is an accepted formula.

In what way is it decreasing?

In the way of „I can’t say by set inclusion with a straight face”.

One by one. Every endsegment transforms a natural number from content
inside to index outside. That is guaranteed by mathematics:
∀k ∈ ℕ : ∩{E(0), E(1), E(2), ..., E(k+1)} = ∩{E(0), E(1), E(2), ...,
E(k)} \ {k+1}. Note the universal quantifier.

 

Except that you don't understand that infinitiy minus 1 is still
infinity, and thus did not "decrease".

 
The sets E(n) decrease. If the sequence (E(n)) could not get empty one
by one then Cantor could not set up an infinite sequence using all
indices n of that sequence.

The „sequence” doesn’t „get” empty. No element is empty.

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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.