Re: Incompleteness of Cantor's enumeration of the rational numbers (extra-ordinary)

On 1/14/25 3:47 AM, WM wrote:

On 14.01.2025 00:47, Richard Damon wrote:

On 1/13/25 7:42 AM, WM wrote:

On 13.01.2025 03:54, Richard Damon wrote:

On 1/12/25 5:58 AM, WM wrote:

On 11.01.2025 14:34, joes wrote:
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1, 2, 3, 4, 5, ..., ω becomes 2, 4, 6, 8, 10, ..., ω, ω+2, ω+4, ..., ω2.

No. There is no x e N such that 2*x >= omega. You have listed two
consecutive infinities on the right.

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There is a basic law: When a sequence of regular distances is multiplied by 2, then a sequence of regular distances results.

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Which isn't applicable, since their isn't such a "regular distance"

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Between all natural numbers, there is a regular distance. When doubled, the new ones do not fit between the old ones and ω because nothing fits between ℕ and ω.

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But omega isn't a natural number, so the space between the ... and omega isn't the same as the space between two consecutive natural numbers.

 Maybe. But the space between the natural numbers is doubled while the accessible space remains constant in actual infinity.

Yes, and double infinity is still just infinity.

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And the distance between the ... and omega *IS* big enough to fit the doubling of all the numbers (without needing to make any new ones).

 If there is space, then it is filled before doubling already. Why shouldn't it?

No, the numbers are of a different class, and the space is inherent in the classes.
The numbers in the ... are all finite (but an infinite number of them) omega is infinite (countably infinite). There is always an infinite space between the infinite set of Natural Numbers and that first Countably infinite number.

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your starting premise is just

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to double all numbers which fit between 1 and ω.

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Which are still just the even numbers between 1 and omega.

 ∀n ∈ ℕ: 2n > n. All numbers are doubled. Their number remains the same (not only the cardinality, but the reality). Half of all are deleted. Half are new.

So? Double and Half of infinity is still infinity.
You are just stuck on our false lies from your finite thinking.

 

correct understanding that all between 1 and ω is available for doubling and nothing can be inserted.

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And nothing NEEDS to be inserted.

 It cannot. Therefore half of the results is larger than ω.

YOu *THINK* it can not, because your brain is to small to understand.
Therefore, you have invented out of your own imagination the lie of "dark numbers" to try to explain what you don't understand.
All that does is prove you don't understand what you are talking about.
Your stupidity is so great, you just can not imagine things being other than what you think they are, which is the highest form of ignorance.

 Regards, WM