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From: Richard Damon <richard@damon-family.org>
Newsgroups: sci.logic
Subject: Re: Replacement of Cardinality
Date: Sun, 28 Jul 2024 13:07:59 -0400
Organization: i2pn2 (i2pn.org)
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On 7/28/24 7:55 AM, WM wrote:
> 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}

And who was using that set?

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

But what number became ω when doubled?

Every natural number when doubled is a Natural Number.

so the result should be { 2, 4, 6, ... 2*ω}.

>>
>> 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}.

But why? we were talking about the infinite set of the Naturals.

>>
>> 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.

Which isn't a Natural Number, as if it was, then that set of Natural 
Numbers would have a maximun member and be finite.

Note, ω-1 doesn't exist in the base transfinite numbers, just as -1 
doesn't exist in the Natural Numbers, you can't go below the first element.

But, just as we can expand the Natural Numbers to the Integers, and get 
negative numbers, we also might be able to define an extention to the 
transfinite numbers that can have a ω-1 element.

>>
>> 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.

And thus you can't have ω-1, just like you can't have -1 in the Natural 
Numbers.

>>
> The fact that you can't understand this is deplorable but does not make 
> my theory wrong.

The fact that your theory is inconsistant makes it 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.
> 

Nope, since that implies there is a highest Natural Number, which breaks 
their definition,

It just shows you mind can't handle proper logic of unbounded sets, but 
is stuck with bounded logic that just breaks when used with unbounded sets.

> Regards, WM
> 
> 
>