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From: "Chris M. Thomasson" <chris.m.thomasson.1@gmail.com>
Newsgroups: sci.math
Subject: Re: how
Date: Fri, 12 Apr 2024 13:16:54 -0700
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On 4/12/2024 9:40 AM, WM wrote:
> Le 12/04/2024 à 17:51, Tom Bola a écrit :
>> WM schrieb:
>>
>>> Le 12/04/2024 à 17:00, Tom Bola a écrit :
>>>> WM schrieb:
>>>>
>>>>> Le 12/04/2024 à 16:40, Tom Bola a écrit :
>>>>>> WM schrieb:
>>>>>>
>>>>>>> Le 12/04/2024 à 15:56, Tom Bola a écrit :
>>>>>>>> WM schrieb:
>>>>>>>
>>>>>>>>> Consider the set {1, 2, 3, ..., ω} and multiply every element 
>>>>>>>>> by 2 with the result {2, 4, 6, ..., ω*2}. What elements fall 
>>>>>>>>> between ω and ω*2? 
>>>>>>>>
>>>>>>>> {w+1, w+2, w+3, ...} 
>>>>>>>
>>>>>>> No, all elements emergeing from doubling have larger distances 
>>>>>>> than 1.
>>>>>>>>
>>>>>>>>> What size has the interval between N*2 and ω*2? 
>>>>>>>>
>>>>>>>> N*2 is not a number, so there is no interval between it and w*2
>>>>>>>
>>>>>>> N*2 is a set having elements but not including w*2. So there is a 
>>>>>>> distance.
>>>>>>
>>>>>> This is wrong because there is a distance to any element of that 
>>>>>> set. But you probably are meaning the distance between the set 
>>>>>> limit of IN which is w and w*2 
>>>>>
>>>>> I am meaning the distance between N*2 and ω*2 after multiplication. 
>>>>
>>>> Yes, that is the set after multiplication: {0, 2, 4, 6, ..., w, w+1, 
>>>> w+2, w+3, ..., w*2} 
>>>
>>> Why are the distances below ω 2 but beyond ω 1?
>>
>> This is the union of the image from IN under f(n)=2n and the "elements 
>> fall between ω and ω*2" that you wanted above
> 
> I wanted the image of 1, 2, 3, ..., ω under multiplication by 2.
>>
>> The image of IN under f(n)=2n and w is still {0, 2, 4, 6, ..., w*2}
> 
> Yes, but where is ω in this sequence?


set A [ 2, 4, 6, 8, ... ]
set B [ 1, 3, 5, 7, ... ]
set C [ 1, 2, 3, 4, ... ]

All three sets are infinite. So, the "size" of each set is infinite.