Re: The non-existence of "dark numbers"

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.

 It thus has
no least member.

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

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

Bijections have no limit.
"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.
"such that every element of the set stands at a definite position of this sequence" [E. Zermelo: "Georg Cantor – Gesammelte Abhandlungen mathematischen und philosophischen Inhalts", Springer, Berlin (1932) p. 152]

 In the limit, it passes
_all_ places.

Do you think that Cantor's above explanations are wrong?
   In informal language, it "disappears off to infinity",
There is no chance to disappear. And never infinity is reached.

and thus is no longer at one of the numbered places.

"such that every element of the set stands at a definite position of this sequence" [E. Zermelo: "Georg Cantor – Gesammelte Abhandlungen mathematischen und philosophischen Inhalts", Springer, Berlin (1932) p. 152]

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?
Regards, WM