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From: "Chris M. Thomasson" <chris.m.thomasson.1@gmail.com>
Newsgroups: sci.math
Subject: Re: how
Date: Sat, 13 Apr 2024 12:12:48 -0700
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On 4/12/2024 3:22 PM, FromTheRafters wrote:
> Chris M. Thomasson has brought this to us :
>> 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.
> 
> In this case, yes, since they are all countable. BTW, sets should be in 
> curly brackets not square ones.

Damn! I am just way to used to arrays in C/C++ so I tend to use the 
square brackets.