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

Am Sat, 14 Dec 2024 17:00:43 +0100 schrieb WM:

That pairs the elements of D with the elements of ℕ. Alas, it can be
proved that for every interval [1, n] the deficit of hats amounts to
at least 90 %. And beyond all n, there are no further hats.

But we aren't dealing with intervals of [1, n] but of the full set.

Those who try to forbid the detailed analysis are dishonest swindlers
and tricksters and not worth to participate in scientific discussion.

No, we are not forbiding "detailed" analysis

Then deal with all infinitely many intervals [1, n].

??? The bijection is not finite.

 

The problem is that you can't GET to "beyond all n" in the pairing,
as there are always more n to get to.

If this is impossible, then also Cantor cannot use all n.

Why can't he? The problem is in the space of the full set, not the
finite sub sets.

The intervals [1, n] cover the full set.

Only in the limit.

 

Yes, there are only 1/10th as many Black Hats as White Hats, but
since that number is Aleph_0/10, which just happens to also equal
Aleph_0, there is no "deficit" in the set of Natual Numbers.

This example proves that aleph_0 is nonsense.

Nope, it proves it is incompatible with finite logic.

There is no other logic.

There is the logic of the infinite.