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

On 17.01.2025 01:56, Jim Burns wrote:

On 1/16/2025 8:32 AM, WM wrote:

A potentiallyᵂᴹ infiniteˢᵉᵗ set
has all available places occupied,

>
But it can grow.

 No, it can't.

Mimicking Damon?
"Potential infinity refers to a procedure that gets closer and closer to, but never quite reaches, an infinite end. For instance, the sequence of numbers 1, 2, 3, 4, ... gets higher and higher, but it has no end; it never gets to infinity.[...]
Completed infinity, or actual infinity, is an infinity that one actually reaches; the process is already done. For instance, let's put braces around that sequence mentioned earlier: {1, 2, 3, 4, ...}. With this notation, we are indicating the set of all positive integers." [E. Schechter: "Potential versus completed infinity: Its history and controversy" (5 Dec 2009)]
Note: "With this notation, we are indicating the set of all positive integers." That implies, with the other notion we are not.

 

Multiplying all its elements by 2
creates new elements.

 No. it doesn't.

If all are there and all are doubled, then greater elements are produced. This is the mathematics of 2n > n. That you cannot accept it, shows that your set theory contradicts mathematics.

 

the same as any other set,
which is to say,
it is (has been, will be) completeᵂᴹ.

>
Then new numbers must be outside.

 There are no new numbers.

2n > n proves new  and greater numbers than are doubled.

 A step is never from finite to infinite.
Therefore, a step never crosses ω

Therefore no ω does exist. That is the only alternative. Potential infinity.
Regards, WM