Re: Replacement of Cardinality

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 :
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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.

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That is true in potential infinity. But I assume actual infinity.

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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