Re: The non-existence of "dark numbers"

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

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.

That's your lack of understanding of things infinite.

 It thus has no least member.

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

Eh?  Members of what?  After what?  Your connection with the thread of
discussion has become very tenuous.

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

Bijections have no limit.

That has no connection with what I wrote.  Sequences and series may have
limits, not bijections.  What we're talking about is a sequence of
positions the distinguished element is at.  This is a sequence of natural
numbers.  At step n, the element is at position n.  After an "infinite
number of steps", the distinguished element is not at a naturally
numbered position - it has "disappeared".

"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)]

Irrelevant.

Limits are not determined places.

Meaningless.

"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]

Irrelevant.

In the limit, it passes _all_ places.

Do you think that Cantor's above explanations are wrong?

I think Cantor would have and did understand the current situation.  What
you have quoted from Cantor are not explanations of what we are
discussing.

  In informal language, it "disappears off to infinity",

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

Tell us all, then, at which element it ends up at.  And none of this
nonsense about "dark numbers", please.

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]

Irrelevant.

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?

That has no connection or relevance to my point, which you have evaded
addressing.  I don't think Cantor was wrong, in general.  But I do think
you base your mathematical thinking on faulty intuition, and not on the
axioms and logic which have chrystallised out of a lot of very clever
thinking over the last few centuries.

Regards, WM

--
Alan Mackenzie (Nuremberg, Germany).