how

On 4/7/24 9:23 AM, WM wrote:

Le 07/04/2024 à 13:16, Richard Damon a écrit :

On 4/7/24 4:32 AM, WM wrote:

Le 06/04/2024 à 22:03, Richard Damon a écrit :

On 4/6/24 3:40 PM, WM wrote:

Le 06/04/2024 à 15:58, Richard Damon a écrit :

On 4/6/24 9:55 AM, WM wrote:

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That mapping is Cantor's proposal. But for every other mapping, the O's would also remain. All O's! It is th lossless exchange which proves it.

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Cantor's proposal is between members of two distinct sets.

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No. He does not specify that. And there is no reason to do so, except that it can be used to contradict the ridiculous nonsense that there are as many fractions as prime numbers.y

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But he DOES, as he talks about the two SETS of numbers that are matched up.

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One set and its subset. Dedekind: A system S is said to be /infinite/ if it is similar to a real part of itself. To consider them as two sets does not change the numbers of elements.

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But does affect your logic of pairing.

 No. Since there are precisely as many natnumbers n as natnumber fractions n/1, nothing is affected. The only effect is that the Os can be proven to remain the same number in every step. This is true in all mappings but more easily seen in mine.

Yes, the size of the Natual Numbers is the same size as the Size of the Rational Fractions that have a denomenator equal to 1.
This doesn't mean it can't ALSO be the same size as the full field of n/d.
Your inability to understand that is just your own problem.

 

So, With infinite sets, a proper subset CAN be the same size as its parent.

 Impossible.

Nope, PROVEN.
Since the DEFINITION of "Same Size" is the ability to make a 1-to-1 mapping between the sets.
Do you want to claim that two sets that you can match EVERY DISTINCT element of one to a UNIQUE DISTINCT ELEMENT of the other are NOT the same size?
and we can build such a mapping between the set of natural Numbers (N) with the set of even Numbers (E).
Since for ALL elements n, a member of the Natural Numbers, there exists an element e, a member of tghe Even Nubers, such that the value of e is twice the value of n (e = 2n)
EVERY element of N is mapped to a DISTINCT element of E.
Try to find an exception.
If E is smaller than N, then BY DEFINITION, when building a bijection, two different values of n must map to the same value of E, but that never happens.
But E is ALSO a proper subset of the Natural Numbers, as we can also build E by removing all the "odd" values from the Natural Numbers.
In fact, it turns out that the size of the set of Natural Numbers is exactly the same size of ANY non-finite subset of any set based on some bounded length tuple of natural numbers (like the rationals). They are all "Countably Infinite" with a size of Aleph_0.

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You are just PROVING you don't understand how infinity works,

 I understand that a crowd of fools has been tricked by Cantor.

NOPE, you have FOOLED YOURSELF by beleiving your own lies.
Your problem is your instance on using "finite" logic on infinite sets, which makes your world blow itself up in contradictions.
Which seems to have blown your brains out of your head.

 Regards, WM