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From: "Chris M. Thomasson" <chris.m.thomasson.1@gmail.com>
Newsgroups: sci.math
Subject: Re: How many different unit fractions are lessorequal than all unit
 fractions? (infinitary)
Date: Sun, 20 Oct 2024 12:59:50 -0700
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On 10/20/2024 12:54 PM, WM wrote:
> 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}.


2 * any_natural_number = a_natural_number

Got it?