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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 21:54:39 +0200
Organization: A noiseless patient Spider
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On 20.10.2024 21:42, Jim Burns wrote:
> On 10/20/2024 3:24 PM, WM wrote:
>> On 20.10.2024 20:20, Jim Burns wrote:
>>> On 10/20/2024 3:48 AM, WM wrote:
>>>> On 20.10.2024 00:54, Jim Burns wrote:
>>>>> On 10/19/2024 2:19 PM, WM wrote:
> 
>>>>> A doubled finite is finite.
>>>>
>>>> If all finites are doubled,
>>>> then not all results can be in that set.
>>>
>>> If all finites are doubled,
>>> then all results are in the set of finites.
>>
>> But not in the mapped or multiplied range.
> 
> If n is countable.to from 0
> then ⟨0,1,...,n-1,n⟩ exists
> 
> If ⟨0,1,...,n-1,n⟩ exists
> then ⟨n,n+1,...,2⋅n-1,2⋅n⟩ exists
> 
> If ⟨0,1,...,n-1,n⟩ and ⟨n,n+1,...,2⋅n-1,2⋅n⟩ exist
> then ⟨0,1,...,n-1,n,n+1,...,2⋅n-1,2⋅n⟩ exists
> 
> If ⟨0,1,...,n-1,n,n+1,...,2⋅n-1,2⋅n⟩ exists
> then 2⋅n is countable.to from 0
> 
> ⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯
> If n is countable.to from 0
> then 2⋅n is countable.to from 0
> 
>>> If all finites are doubled,
>>> then all results are in the set of finites.
>>
>> But not in the mapped or multiplied range.
> 
> If n is countable.to from 0
> then 2⋅n is countable.to from 0

Nevertheless 2n is not in the set {1, ..., n}.

Regards, WM