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

On 13.01.2025 21:44, Jim Burns wrote:

On 1/13/2025 12:29 PM, WM wrote:

On 13.01.2025 18:06, Jim Burns wrote:

On 1/13/2025 7:48 AM, WM wrote:

ℕ is only invariable in the sense which we use.

That is potential infinity.

However,
you (WM) are convinced that
a set (such as ℕ) larger than
any set with sets.different.in.size.by.one
changes (has elements inserted or deleted)
in order to not.change.in.size.by.one.

In actual infinity all elements are invariable.

>
 

⎜ #⟦0,𝔑⦆ = #ℕ
⎜
⎜ ⟦0,𝔑+1⦆ is 'fixed', too.

>
ω

 Is ω = ⟦0,𝔑+1⦆ your 'fixed' (our 'finite')?

ω is the first infinite ordinal by definition.

What about ω+1 and ω+2?

Infinite ordinals too.

 

⎜ ⟦0,𝔑+1⦆ ⊆ ℕ

>
No.

 Yes.

No.

∀𝔑 ∈ ℕ: #⟦0,𝔑⦆ < #⟦0,𝔑+1⦆ ≤ #ℕ

No. The sequence of endsegments can get empty by
∀k ∈ ℕ : E(k+1) = E(k) \ {k+1} . There is a sharp end.
Regards, WM