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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: Thu, 24 Oct 2024 13:58:38 +0200
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On 24.10.2024 13:40, Richard Damon wrote:
> On 10/24/24 7:07 AM, WM wrote:
>> On 24.10.2024 05:04, Richard Damon wrote:
>>> On 10/23/24 10:46 AM, WM wrote:
>>>> On 23.10.2024 13:37, Richard Damon wrote:
>>>>> On 10/22/24 12:12 PM, WM wrote:
>>>>>> On 22.10.2024 18:03, Jim Burns wrote:
>>>>
>>>>>>> ∀n ∈ ℕ:  2×n ∈ ℕ
>>>>>>
>>>>>> Not if all elements are existing before multiplication already.
>>>>>
>>>>> IF not, then your actual infinity wasn't actually infinite
>>>>
>>>> It is infinite like the fractions between 0 and 1. When doubling we 
>>>> get even-numerator fractions, some of which greater the 1.
>>>
>>> But from 0 to 1 isn't an infinite distance
>>
>> Measured in rational points it is infinite.
> 
> But values are not measured in "rational points".

We can measure in rational points. Between two such points there are 
many dark ones, but that is the same between 0 and omega.
> 
>> It is claimed that there are all numbers. "That we have for instance 
>> when we consider the entirety of the numbers 1, 2, 3, 4, ... itself as 
>> a completed unit, or the points of a line as an entirety of things 
>> which is completely available. That sort of infinity is named actual 
>> infinite." [D. Hilbert: "Über das Unendliche", Mathematische Annalen 
>> 95 (1925) p. 167]
> 
> So?
> 
> If you HAVE all the numberes, 1, 2, 3, 4, ... that set goes on FOREVER 
> and doesn't have a upper end.

But it has all. All can be doubled.
> 
> If you have the COMPLETE unit, it doesn't have a highest number.

But it has all. All can be doubled.
> 
> Your operation of doubling the values on the line from 0 to 1 isn't 
> operating the property that that set is infinite on, so doesn't follow 
> the law of the infinite.

Infinite sets can be mapped completely, according to set theory.

>>> There is nothing about being complete that means it needs to have an 
>>> "end"
>>
>> Whatever, it is complete and all its numbers can be doubled. Some are 
>> resulting in larger numbers than have been doubled.
> 
> Nope, as every number (A Natural Number) doubles to another number in 
> that set (The Natural Numbers) so you never left the set.

I take all of it.

Regards, WM