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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, 15 Oct 2024 22:41:08 -0400
Organization: i2pn2 (i2pn.org)
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On 10/14/24 6:35 AM, WM wrote:
> On 12.10.2024 22:47, Richard Damon wrote:
>> On 10/12/24 2:39 PM, WM wrote:
>>> On 11.10.2024 20:02, Jim Burns wrote:
>>>
>>>> "Potential ℕ" vs. "actual ℕ" leaves unchanged
>>>> which claims we reason to.
>>>
>>> Potential infinity: All natural numbers when double yield natural 
>>> numbers but larger numbers are among the result.
>>> In actual infinity no new naturals can be created but since doubling 
>>> doubles the value of each number, numbers of the second number class 
>>> are created.
> 
>> No, in actual infinity no new natural can be created,
> 
> True.
> 
>> but every natural number can find the one that is twice itself in the 
>> set,
> 
> False. n*2 > n for all numbers, even for infinite numbers.
> 
> Regards, WM

Nope, that is a property of finite numbers, and infinite ordinals only. 
It doesn't apply for infinite cardinal numbers. (Omega * 2 is bigger 
than Omega, but Aleph_0 * 2 is just Aleph_0)

For finite numbers n*2 is also finite, and thus less than Omega, and in 
the actually infinite set, that doesn't require the creation of new 
numbers, as the full infinite set was there to begin with, even if you 
didn't notice them when you looked. (And that doesn't make them "dark")