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From: WM <wolfgang.mueckenheim@tha.de>
Newsgroups: sci.math
Subject: Re: The non-existence of "dark numbers"
Date: Fri, 4 Apr 2025 14:45:49 +0200
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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