Re: What is the interval between ℕ and ω when doubled?

On 4/15/24 7:36 AM, WM wrote:

Le 14/04/2024 à 21:25, Richard Damon a écrit :

On 4/14/24 2:08 PM, WM wrote:

Le 13/04/2024 à 16:48, Mikko a écrit :

On 2024-04-13 12:13:07 +0000, WM said:
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\omega is not an element of |N.

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That is true. The question concerns the distance between both.
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The second set does not contain \omega.

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But it contains ω*2.

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What size has the interval from sweet to blue?

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Are they points on the ordinal axis?

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No, sweet, blue, and ℕ are not points on the ordinal axis.

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But ω and all elements of ℕ are points on the ordinal axis.

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ω only exist on that TRANSFINITE ordinal axis, not the finite ordinal axis.

 Some ordinal numbers of the beginning of the sequence (with k, m, n  ) are:
 0, 1, 2, 3, ..., ,  + 1, ...,  + k, ...,  +  (= 2), 2 + 1, ..., k, ..., k + m, ...,  (= 2), 2 + 1, ..., 2 + , ..., 2 + k + m, ..., 22, ..., 2k + m + n, ..., 3 + 2k + m + n, ..., k, .., ,  + 1, ..., k, ..., +1, +1 + 1, .., k, ..., 2, ..., , ...,  (= 0), 0 + 1, ..., 00, ..., 000, ..., 000 (= 1), 1 + 1, ..., 111 (= 2), ..., 1, ... .
 Better readable in Transfinity, https://www.hs-augsburg.de/~mueckenh/Transfinity/Transfinity/pdf, p.42.
 Regards, WM
 

Just shows you don't understand what you are talking about.
For instance, the representation of all pairs of natural numbers is ω^2, not 2^w.
Cantor shows that w^2 is in the same size class as ω, but 2^ω is in a higher size class.
Of course, since you logic can't handle infinite sets, all the "contradictions" you try to point out are just proofs that you logic can't handle the sets, not that there is something inherently wrong with Cantor's argument. (You just reject the base logic he presumes to be using).