Re: The non-existence of "dark numbers"

Am Wed, 02 Apr 2025 16:07:20 +0200 schrieb WM:

On 27.03.2025 22:34, Alan Mackenzie wrote:

WM <invalid@no.org> wrote:

Am 26.03.2025 um 22:34 schrieb Alan Mackenzie:

WM <wolfgang.mueckenheim@tha.de> wrote:

On 26.03.2025 07:24, Jim Burns wrote:

"WM-logic" says that lossless exchanges at finite steps are lossless
....

Nobody has contradicted this, that I'm aware of.  It's the
accumulation of _all_ of these lossless exchanges where unexpected
things happen.

If they happen, then there is a first instance where they happen.
Every n.e. subset of a countable set has a first element.

The set of integer steps at which a loss occurs is empty.

There are no other steps at which anything could occur.

There is the limit.

 It thus has no least member.

Nevertheless all members are finite integers, and afterwards nothing
happens anymore.

After what exactly?

It is only in the infinite limit where the loss occurs.

Bijections have no limit.

Good that you're finally coming around on this.

"The infinite sequence thus defined has the peculiar property to contain
the positive rational numbers completely, and each of them only once at
a determined place." [G. Cantor, letter to R. Lipschitz (19 Nov 1883)]
Limits are not determined places.

Why should you think "there" is a rational number?

In the limit, it passes _all_ places.
  In informal language, it "disappears off to infinity",

There is no chance to disappear. And never infinity is reached.

"Disappear" is just a figure of speech, like running out of view.
Infinity is "reached" in - wait for it - the limit.

and thus is no longer at one of the numbered places.
You base your mathematical thinking on faulty intuition.  You do not
base it on the axioms and logic which have chrystallised out of a lot
of very clever thinking over the last few centuries.

Do you think that Cantor was wrong?

Aren't you the one?

--
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.