Re: Contradiction of bijections as a measure for infinite sets

Le 07/04/2024 à 21:47, Jim Burns a écrit :

On 4/6/2024 3:49 PM, WM wrote:

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

 ...your anti.arithmetism.

If all products of the sequence 1, 2, 3, ..., remain below ω but ω ss mapped to ω2, then the distance between ℕ and ω is increased from 0 to infinity by multiplying.

>
The successor operation is closed in
the natural numbers.

For visible numbers only.

Natural.number.addition is closed in
the natural numbers.
Natural.number.multiplication is closed in
the natural numbers.
 Those claims are theorems.
In order to be theorems,
it is important to know the meaning of
successor, natural number, addition and multiplication.
 If you want to see the proofs, ask.

The proofs are valid for visible numbers only. But most numbers are invisible as is shown by the fact that no numbers fits between ℕ and ω.

Two times all numbers of 1, 2, 3, .., ω
will double the distance between ℕ and ω
to that between 2ℕ and 2ω.

 No, it won't.

*2 doubles every structure. According to my logic.
Regards, WM