Consider the set {1, 2, 3, ..., ω} and multiply every element by 2 with
the result {2, 4, 6, ..., ω*2}. What elements fall between ω and ω*2?
What size has the interval between ℕ*2 and ω*2?

Regards, WM

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Am Sun, 07 Apr 2024 08:38:56 +0000 schrieb WM:

Consider the set {1, 2, 3, ..., ω} and multiply every element by 2 with
the result {2, 4, 6, ..., ω*2}. What elements fall between ω and ω*2?
What size has the interval between ℕ*2 and ω*2?

Regards, WM

\omega is not an element of |N. The second set does not contain \omega.
-- ajh

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Le 11/04/2024 à 10:15, [email protected] a écrit :

Am Sun, 07 Apr 2024 08:38:56 +0000 schrieb WM:

Consider the set {1, 2, 3, ..., ω} and multiply every element by 2 with
the result {2, 4, 6, ..., ω*2}. What elements fall between ω and ω*2?
What size has the interval between ℕ*2 and ω*2?

\omega is not an element of |N.

That is true. The question concerns the distance between both.

The second set does not contain \omega.

But it contains ω*2.

Regards, WM

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On 2024-04-11 13:49:35 +0000, WM said:

Le 11/04/2024 à 10:15, [email protected] a écrit :

Am Sun, 07 Apr 2024 08:38:56 +0000 schrieb WM:

Consider the set {1, 2, 3, ..., ω} and multiply every element by 2 with >>> the result {2, 4, 6, ..., ω*2}. What elements fall between ω and ω*2? >>> What size has the interval between ℕ*2 and ω*2?

\omega is not an element of |N.

That is true. The question concerns the distance between both.

The second set does not contain \omega.

But it contains ω*2.

What size has the interval from sweet to blue?

--
Mikko

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Le 12/04/2024 à 09:51, Mikko a écrit :

On 2024-04-11 13:49:35 +0000, WM said:

Le 11/04/2024 à 10:15, [email protected] a écrit :

Am Sun, 07 Apr 2024 08:38:56 +0000 schrieb WM:

Consider the set {1, 2, 3, ..., ω} and multiply every element by 2 with >>>> the result {2, 4, 6, ..., ω*2}. What elements fall between ω and ω*2? >>>> What size has the interval between ℕ*2 and ω*2?

\omega is not an element of |N.

That is true. The question concerns the distance between both.

The second set does not contain \omega.

But it contains ω*2.

What size has the interval from sweet to blue?

Are they points on the ordinal axis?

Regards, WM

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On 2024-04-13 12:13:07 +0000, WM said:

Le 12/04/2024 à 09:51, Mikko a écrit :

On 2024-04-11 13:49:35 +0000, WM said:

Le 11/04/2024 à 10:15, [email protected] a écrit :

Am Sun, 07 Apr 2024 08:38:56 +0000 schrieb WM:

Consider the set {1, 2, 3, ..., ω} and multiply every element by 2 with >>>>> the result {2, 4, 6, ..., ω*2}. What elements fall between ω and ω*2? >>>>> What size has the interval between ℕ*2 and ω*2?

\omega is not an element of |N.

That is true. The question concerns the distance between both.

The second set does not contain \omega.

But it contains ω*2.

What size has the interval from sweet to blue?

Are they points on the ordinal axis?

Regards, WM

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

--
Mikko

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

On 2024-04-13 12:13:07 +0000, WM said:

\omega is not an element of |N.

That is true. The question concerns the distance between both.

The second set does not contain \omega.

But it contains ω*2.

What size has the interval from sweet to blue?

Are they points on the ordinal axis?

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

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

Regards, WM

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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:

\omega is not an element of |N.

That is true. The question concerns the distance between both.

The second set does not contain \omega.

But it contains ω*2.

What size has the interval from sweet to blue?

Are they points on the ordinal axis?

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

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

Regards, WM

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

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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:

\omega is not an element of |N.

That is true. The question concerns the distance between both.

The second set does not contain \omega.

But it contains ω*2.

What size has the interval from sweet to blue?

Are they points on the ordinal axis?

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

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

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

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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:

\omega is not an element of |N.

That is true. The question concerns the distance between both.

The second set does not contain \omega.

But it contains ω*2.

What size has the interval from sweet to blue?

Are they points on the ordinal axis?

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

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

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).

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