Re: Incompleteness of Cantor's enumeration of the rational numbers

On 11/25/24 7:18 AM, WM wrote:

On 24.11.2024 20:14, Richard Damon wrote:

On 11/24/24 12:32 PM, WM wrote:

 

The finite sets contain all hats because all natural numbers and all 10n are in finite sets. No hat is outside.>

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But there is no finite set with ALL natural numbers.

 ℕ is fixed, that means |ℕ| is fixed.

But infinite, and thus a series of finite sets are not actually an approximation for it.
Your problem is that your "generator" for the Natural Numbers never finishes, so you never get to have them.

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Like usual, you mess up with your qualifiers.

Limit theory only works if the limit actually exists

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If limits exist at all, then the limit of the sequence 1/10, 1/10, 1/10, ... does exist.

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But the concept of 1/10th of an infinte set does not exist..

 It does.

Nope, not the way you want it to be. 1/10 of Aleph_0 is still Aleph_0 so the two sets are the same size.

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You can get things that APPEAR to reach a limit, but actually don't.

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But if infinite sets do exist, then the set ℕ does exist, and all its elements are members of finite intervals (0, n].

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No, any given element is a member of a finite set, but you can't then say that ALL are in such a set.

 All are in the union of all finite sets.

But your logic can't handle an infinite union, as it can never complete the process.

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WIth infinity, Any and All are different qualifiers.

 Use them to increase your qualification.

Why, I understand the differences. You seem to not, and thus make your errors and have exploded your mind with the contradictions.

 Regards, WM