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

On 18.01.2025 14:46, Richard Damon wrote:

On 1/17/25 2:34 PM, WM wrote:

Am 17.01.2025 um 15:52 schrieb Richard Damon:

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

>

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,

All which are smaller than the greatest are useless. That should even be understood by a very limited brain. There is no greatest. Therefore all are useless.

So, you agree that your logic makes everything worthless.

It makes nonsense worthless. I can prove by induction that every FISON can be omitted because it is useless.

THus, your false premise that without a biggest, the smaller are worthless is just incorrect, and thus your conclusion is incorrect.

Name any FISON that is required to produce ℕ by a union of FISONs.
The set of required FISONs is well defined because for every FISON we can decide whether it is required in the union.
Further, according to set theory, every well defined set of ordinal numbers has a first element. The FISONs F(n) must obey this theorem because they can be ordered by their greates ordinals n.
You simply violate this fact.

 It may be that without being able to have a biggest Natural Number you think that it makes that set worthless, but that is just your error.

The error is your handwaving claim that infinitely many FISONs are required. Infinitely many failures will not yield a success.
Regards, WM