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From: WM <wolfgang.mueckenheim@tha.de>
Newsgroups: sci.math
Subject: Re: How many different unit fractions are lessorequal than all unit
 fractions? (infinitary)
Date: Sun, 20 Oct 2024 11:59:31 +0200
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On 20.10.2024 11:39, Alan Mackenzie wrote:
> WM <wolfgang.mueckenheim@tha.de> wrote:
>> On 19.10.2024 22:19, Alan Mackenzie wrote:
>>> WM <wolfgang.mueckenheim@tha.de> wrote:
> 
>>>> The result covers twice the interval, ....
> 
>>> It does not, except in the sense that twice infinite = infinite.
> 
>> Twice the infinite contains numbers not in the first infinite.
> 
> It does not.

2n > n holds for all natural numbers.
> 
>> What else could twice mean?
> 
> It doesn't really mean much at all.

It means that after doubling more numbers are there than have been doubled.

>> These numbers are dark. Their existence is proven by the fact that
>> ∀n ∈ ℕ, n < 2n.
> 
> Their non-existence I proved in this thread, many posts ago.

Based on wrong assumptions.

>> The only alternative is that doubling numbers creates only doubled
>> numbers.
> 
> That's more or less what happens, except that numbers aren't "created".
> They just are.

That's not what happens i mathematics.
> 
>> It is not acceptable.
> 
> It is accepted by mathematicians.

Only by those who never thought about that topic.
> 
>> It cannot be apologized by the possibility to map all numbers on even
>> numbers.
> 
> There exists a 1-1 correspondence between all natural numbers and all
> even natural numbers, a proper subset of all natural numbers.  The set of
> all natural numbers is thus infinite.

The mapping needs larger numbers than have been mapped. That is not a 
matter of infinite sets but basic mathematics of all natural numbers. 
The correspondence between all natural numbers and all even natural 
numbers requires a variable "all". Potential infinity.

Regards, WM