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

On 1/17/25 8:00 AM, WM wrote:

On 17.01.2025 12:18, joes wrote:

Am Fri, 17 Jan 2025 11:39:55 +0100 schrieb WM:

 

Logic says that every FISON which is smaller than another FISON can be
dropped. Logic says even more: Every FISON which is smaller than ℕ can
be dropped. Therefore all can be dropped.
Do you know a FISON that is smaller than ℕ but cannot be dropped?

"Logic" (by which you mean common sense) does not say you can drop all
segments, only finitely many.

 By logic I mean logic. Since the small numbers are always covered by the greater FIS=ONs, the smaller FISONs can be dropped. Either the complete set ℕ is produced by one FISON, which then is not finite and therefore is not a FISON, or it is not produced by FISONs.

It is produced by the INFINITE set of FISONs,

 This can also be proved by induction: Every FISON is missing some natural numbers and is therefore irrelevant for the union.

Which just shows that any one FISON is insufficient, not that the complete set is.
All your induction proves is that every FISON if finite, not that the set of Natural Numbers can be built by the union of ALL FISONs.

 Regards, WM