Re: Replacement of Cardinality

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

Why?
Note, ω is NOT a member of the Natural Numbers, it is just the "least upper bound" that isn't in the set.
That seems to be your problem in understanding.
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.
Your "logic" just seems to be that ω is just some very big, an perhaps unexpressed, value of a Natural Number, because you "logic" can't actually handle infinite values.
The fact that you can't understand this, doesn't make it not true, just that you own logic is too limited.

Regards, WM