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

Am Thu, 21 Nov 2024 22:40:12 +0100 schrieb WM:

On 21.11.2024 21:59, joes wrote:

Am Thu, 21 Nov 2024 17:55:25 +0100 schrieb WM:

On 21.11.2024 14:05, FromTheRafters wrote:

WM wrote :

>

Let every unit interval after a natural number on the real axis
be coloured white with exception of the powers of 2 which are
coloured black. Is it possible to shift the black intervals so
that the whole real axis becomes black?

The analytical limit is wrong in your opinion?

Which limit?

The limit of the sequence f(n) of relative coverings in the interval
(0, n] is 0, not 1.

Yes, of the infinite real axis none are black.

Not none but less than any definable eps.

Less than every positive number = zero. Also I was pulling your leg.

Which limit?

That is where Cantor's bijections are believed to be complete.

I mean, bijections are also complete for every finite subset.

No, for any finite subset (0, n] there is no complete bijection. Too few
black unit intervals available to cover (0, n].

Haha what I was talking about a function like
{(0, 1), (1, 0), (n, n)} for all n>1.

There IS a bijection between {0, 1, ..., n} and {2^k for all k<n}
for all n e N.

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Am Sat, 20 Jul 2024 12:35:31 +0000 schrieb WM in sci.math:
It is not guaranteed that n+1 exists for every n.