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

On 25.10.2024 16:53, Jim Burns wrote:

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

 

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.

 Correct so far.

>
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 )

 That is wrong in complete infinity.