Re: How many different unit fractions are lessorequal than all unit fractions? (infinitary)

On 10/25/2024 7:42 AM, WM wrote:

Am 25.10.2024 um 06:33 schrieb Jim Burns:

On 10/24/2024 2:58 PM, WM wrote:

The whole set with all its numbers exists
and can be mapped to the double numbers.

>
Yes.
∃⟨0,1,...,n-1,n⟩  ⇒
∃⟨0,1,...,n-1,n,n+1,...,n+n-1,n+n⟩
>

That halves the density
and doubles the covered interval.

>
No.
⟨0,1,...,n-1,n,n+1,...,n+n-1,n+n⟩ is finite.

>
The whole interval (0, ω) is not finite,
let alone the doubled interval.

⟦0,ω⦆ is the set of finite ordinals.
That is the definition of finite ordinal.
That is the definition of ω,
the first ordinal after all finite ordinals.
γ before ω: γ is finite.
γ ∈ ⟦0,ω⦆ ⇒
∀β ∈ ⦅0,γ⟧: ∃α: α+1=β
ω before ξ: ξ is not finite.
ω ∈ ⦅0,ξ⟧ ⇒
¬∀β ∈ ⦅0,ξ⟧: ∃α: α+1=β
(Keep in mind that ¬∃α: α+1=ω )
Any third option, and {ω,ξ} isn't well.ordered.
The "proof" of that is like
the "proof" that 3 is between 2 and 4.
It's what we mean by 'ω'.

If you dislike this result,
then you oppose to complete infinity.

Why would I oppose your (WM's) complete infinity,
merely because it waffles between
pathologically.vagueness and incoherency?
A better question is:
why do you (WM) support it?