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Subject: Re: Replacement of Cardinality
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Date: Sun, 28 Jul 24 11:55:17 +0000
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From: WM <wolfgang.mueckenheim@tha.de>
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Le 27/07/2024 à 14:55, Richard Damon a écrit :
> On 7/27/24 8:16 AM, WM wrote:
>> Le 27/07/2024 à 13:27, Richard Damon a écrit :
>>> On 7/27/24 7:13 AM, WM wrote:
>>>> Le 27/07/2024 à 04:23, Richard Damon a écrit :
>>>>
>>>>> By your logic, if you take a set and replace every element with a 
>>>>> number that is twice that value, it would by the rule of 
>>>>> construction say they must be the same size.
>>>>
>>>> That is true in potential infinity. But I assume actual infinity.
>>>>>
>>>
>>> So, what part is not true?
>> 
>> In potential infinity there is no ω.
>> 
>>> Are you stating that replacing every element with another unique 
>>> distinct element something that make the set change size?
>> 
>> In actual infinity the number of elements of any infinite set is fixed.
>> Doubling all elements of the set ℕ U ω = {2, 4, 6, ..., ω}

Mistake! ℕ U ω = {1, 2, 3, ..., ω}

>> yields the set
>> {2, 4, 6, ..., ω, ω+2, ω+4, ..., ω*2}.
>> 
> 
> Why?

See the correction.
> 
> Note, ω is NOT a member of the Natural Numbers, it is just the "least 
> upper bound" that isn't in the set.

I know. Therefore I wrote ℕ U ω, or better ℕ U {ω}.
> 
> There is no Natural Number that is ω/2 so that doubling it get you to ω, 
> as every Natural Number when doubled gets you another Natural Number.

There is no definable natural number ω/2. But if there are all elements, 
then there is no gap before ω but ω-1.
> 
> Your "logic" just seems to be that ω is just some very big, an perhaps 
> unexpressed, value of a Natural Number,

No, it is the first transfinite number like 0 is the first non-positive 
number.
> 
The fact that you can't understand this is deplorable but does not make my 
theory wrong.
Using the unit fractions itelligent readers understand that there must be 
a first one after zero. Others must believe in the magical appearance of 
infinitely many unit fractions.

Regards, WM