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

On 12/14/24 4:34 PM, WM wrote:

On 14.12.2024 19:53, Richard Damon wrote:

On 12/14/24 10:55 AM, WM wrote:

On 14.12.2024 15:08, Richard Damon wrote:

On 12/14/24 3:53 AM, WM wrote:

>

Please refer to the simplest example I gave you on 2024-11-27:
The possibility of a bijection between the sets ℕ = {1, 2, 3, ...} and D = {10n | n ∈ ℕ} is contradicted because for every interval (0, n] the relative covering is not more than 1/10, and there are no further numbers 10n beyond all natural numbers n. The sequence 1/10, 1/10, 1/10, ... has limit 1/10.

+

Except that we aren't dealng with the FINITE sets of {1, 2, 3, ..., n} but for the full set of { 1, 2, 3, ... }

>
Of course. It consist of all finite n. It is the union of all intervals (0, n]. Not more!

 

But the set of Natural Numbers isn't built that way.

 You ar5e misinformed. The set of natural numbers is the union of all FISONs F(n) = {1, 2, 3, ..., n} = (0, n].
 Regards, WM

 From where did this concept come from?
The Set of Natural Numbers comes from the Generation Algorithm, of the first number, unioned with the successor of each number that was previously created.
We don't union the FISIONS, we make a union of each individual number.
There is no "limit" process, it is just that set.
YOU can't handle this, so you make up your own BROKEN ideas.
Sorry, but is is YOU who is misinformed, because you just reject the sound logical basis of set theory, for your own Naive Mathematics that is just as broken as Naive Set Theory was shown to be, and it doesn't even need as complicated of an example to show that.