Re: Incompleteness of Cantor's enumeration of the rational numbers

Am 16.11.2024 um 23:22 schrieb Moebius:

Am 16.11.2024 um 23:16 schrieb Chris M. Thomasson:

On 11/16/2024 2:11 PM, Moebius wrote:

Am 16.11.2024 um 22:48 schrieb Chris M. Thomasson:

On 11/16/2024 1:29 PM, FromTheRafters wrote:

Chris M. Thomasson pretended:

>
(infinity - infinity) = undefined
>
?

>
https://en.wikipedia.org/wiki/ L%27H%C3%B4pital%27s_rule#Other_indeterminate_forms

>
Or this one: https://en.wikipedia.org/wiki/Indeterminate_form
>
"oo - oo" is an "indeterminate form".
>

>
I must be missing something: [...]
>
[1] = a gallon of water out of an infinite pool
[2] = another gallon of water out of an infinite pool
[3] = on and on... taken to infinity...
>
The pool would always have infinite water for this process?

 Assume that the H2O molecules in the pool are "numerated" by 1, 2, 3, ... (i.e. that ALL H2O molecules in the pool are "numerated" by natural numbers).
 Taking out the "infinite amount of H2O molecules" numerated by 1, 2, 3, ... would lead to an EMPTY pool.

Hint: If the first gallon of water consists of the H2O molecules numbered by 1, ..., n_1, the second gallon of water consists of the H2O molecules numbered by (n_1)+1, ..., n_2 (with n_2 > (n_1)+1), and so on, the "outcome" would be an empty pool. (Hint: try to name the number attached to an H2O molecule which "remains" in the pool.)

On the other hand, taking out the "infinite amount of H2O molecules" numerated by 2, 4, 6, ... would not drain the pool. 😛 It would still be filled (sort of) with infinitely many H20 molecules. 😛
 Be aware if the infinite, man!
 .
.
.