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

On 13.01.2025 20:31, Jim Burns wrote:

On 1/13/2025 12:17 PM, WM wrote:

Doubling of all n
deletes the odd numbers
but cannot change the number of numbers,

 ℕ is the set of finite ordinals.
 There is no finite set larger than ℕ
thus ℕ is infinite.
There is no infinite set smaller than ℕ

The set of prime numbers is smaller than ℕ but appears infinite too because all sets with more than a definable number of numbers appear infinite.

 𝔼 is the set of even finite ordinals.
There is no finite set larger than 𝔼
thus 𝔼 is infinite
𝔼 ⊆ ℕ
#𝔼 ≤ #ℕ
There is no infinite set smaller than ℕ
#𝔼 ≥ #ℕ

That is obviously wrong.

therefore creates even numbers.
They do not fit below ω.

 No.
They fit below ω

In completed infinity all available places are occupied.
∀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.

A step is never from finite to infinite.
Therefore, a step never crosses ω
Therefore, a sum never crosses ω
Therefore, a product never crosses ω
Therefore, a power never crosses ω

All that is true in potential infinity, however it is wrong in completed infinity.
Regards, WM