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From: WM <wolfgang.mueckenheim@tha.de>
Newsgroups: sci.math
Subject: Re: How many different unit fractions are lessorequal than all unit
 fractions? (infinitary)
Date: Fri, 25 Oct 2024 18:57:07 +0200
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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.

> A better question is:
> why do you (WM) support it?

I support it in order to show that your infinity is inconsistent.
Example: Almost all unit fractions cannot be discerned by definable real 
numbers. If they are existing, they are indiscernible, i.e. dark.

Regards, WM
> 
>