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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: Sat, 19 Oct 2024 07:26:20 -0400
Organization: i2pn2 (i2pn.org)
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On 10/19/24 4:23 AM, WM wrote:
> On 18.10.2024 03:26, Richard Damon wrote:
>> On 10/17/24 2:46 PM, WM wrote:
> 
>>> When doubling natural numbers we obtain even numbers which have not 
>>> been doubled.
>>
>> Then your "Actual Infinity" wasn't actually infinte.
>>
>> As it must contain *ALL* the Natural Numbers to be that set.
> 
> Therefore the double numbers are not natural but infinite.

How can doubling a finite number result in an infinite number?

or do you mean are a part of an infinite set, in which case, YES, they 
must be just like the undoubled numbers were.

> 
>>> 2n > n is always true, in finite and in infinite sets.
>>
>> In FINITE or ORDINAL systems, and there 2n will be in the same 
>> actually infinite set as n.
> 
> That is not possible if all natural numbers are doubled. The result 
> covers the interval (0, ω*2) twice as large as the original one (0, ω).

No, since (0, ω) doesn't include ω (and thus just the finite Natural 
numbers), and twice any Natural Number is another finite Natural number, 
the range of the results is the same as input range (0, ω)

>>
>> In Infinte Cardinal spaces (like Aleph_0 is in) 2n == n
> 
> No.

But it is, and saying No doesn't change it.

At best, you can say that you logic doesn't include the Infinte 
Cardinals, but then it has been shown that it doesn't acually have ANY 
of the infinites, as your "Actual Infinity" isn't actually infinite.

> 
> Regards, WM
>