Re: Contradiction of bijections as a measure for infinite sets

Le 06/04/2024 à 17:49, Jim Burns a écrit :

On 4/6/2024 9:44 AM, WM wrote:

But we can use the ordinal axis
as Cantor has described it
0, 1, 2, 3, ...,
ω, ω + 1, ..., ω + k, ...,
ω + ω (= ω2), ω2 + 1, ..
and multiply 0, 1, 2, 3, ..., ω, by 2.

 Two times anything on the first row
is something on the first row.

That is contradicted by this argument: Two times all numbers of 1, 2, 3, .., ω will double the distance between ℕ and ω to that between 2ℕ and 2ω. That distance is less than the distance between ω and ω2. Everything before this is a doubled natural number and simultaneously a transfinite number.
Regards, WM