Re: The non-existence of "dark numbers"

On 03.04.2025 22:05, Alan Mackenzie wrote:

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

table 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 is my lack of believing nonsense.

 

  It thus has no least member.

 

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

 Eh?  Members of what?  After what?

All members n of ℕ enumerating the set of fractions are finite. Nothing is enumerated "in the limit".

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

 

Bijections have no limit.

 That has no connection with what I wrote.

"It is only in the infinite limit where the loss occurs" is pure nonsense.

Sequences and series may have
limits, not bijections.

Therefore your sentence is rubbish. Not indexed fractions don't disappear in the limit.

What we're talking about is a sequence of
positions the distinguished element is at.

We are talking about a claimed bijection. The distinguished element is a not indexed fraction. Fractions don't disappear in the limit.

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".

No, we discuss a bijection with natural numbers. All are fiite. Nothing happens in the limit. Every step is finite.

 

"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.

For you? Not for bijections.

 

Limits are not determined places.

 Meaningless.

For you? Not for bijections.

 

"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.

For you? Not for bijections.

 

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.

They are precisely about what we are discussing, namely a bijection between naturals and fractions.

 

   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.

The infinity of natural numbers never ends but all numbers are finite.

"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.
 

For you? Not for bijections.

That has no connection or relevance to my point, which you have evaded
addressing.

You claimed that not indexed fractions dissappear in the limit. That has been addressed by me and has been qualified as bullshit.
Regards, WM