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From: Richard Damon <richard@damon-family.org>
Newsgroups: sci.math
Subject: Re: How many different unit fractions are lessorequal than all unit
 fractions? (infinitary)
Date: Tue, 8 Oct 2024 08:35:10 -0400
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On 10/8/24 5:54 AM, WM wrote:
> On 07.10.2024 15:19, Alan Mackenzie wrote:
>> WM <wolfgang.mueckenheim@tha.de> wrote:
> 
>>> No, even an unbounded sequence does not get longer when shifted by 
>>> one step.
>>
>> The concept of "length" appropriate for finite sets doesn't apply to
>> infinite sets.
> 
> It is the concept of number of elements. It is appropriate in actual 
> infinity.

Which has been shown to not exist for us finite beings, as it is too big 
for us to see.

> 
>> infinite means "without end" - unendlich.
> 
> Actual infinity means complete. That implies a fixed number.

Which has been shown to not exist for us finite beings, as it is too big 
for us to see.

>>> You don't understand that actual infinity is a fixed quantity.
>>
>> It may be "fixed" whatever that might mean, but to regard it as a
>> "quantity" is more than questionable.
> 
> Fixed means that no element can be added and no element can be lost. The 
> number of nines is fixed. That is an assumption only, but necessary for 
> bijections.

Yes, it is fixed, at INFINITY, which means there is no end to it, and 
thus we can't add a zero "at the end" which doesn't exist.

This is why finite beings can't use "actual infinity" because it is too 
big for us to handle.

> 
>>> Bijection means completeness.
>>
>> No.  Bijection just means a 1-1 correspondence between the elements of
>> two sets. > Nowhere in that definition is any mention of completeness.
> 
> It also means surjectivity. The preimage is complete by definition.

No, surjectivity is something else. Yes, every Bijection is also a 
Surjection (both ways)

>> The diagonal number _is_ complete.  Remember, we are not constructing it
>> tiringly one digit per second, or anything like that.  The entire number
>> is defined and simply exists.
> 
> Only when the set of indexes is complete.

But the completeness is assured due to the induction. We can't actually 
get there ourselves, since we are finite, but we can know it is there.

> 
> Regards, WM
>