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Subject: Re: Replacement of Cardinality
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Date: Sat, 24 Aug 24 14:11:53 +0000
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From: WM <wolfgang.mueckenheim@tha.de>
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Le 24/08/2024 à 00:04, Jim Burns a écrit :
> On 8/22/2024 8:16 AM, WM wrote:
>> Le 21/08/2024 à 20:34, Moebius a écrit :
> 
>>> Yeah, it's like claiming:
>>> "There is an end
>>> (to the natural numbers), 
>>> because at and above omega
>>> there is no natural number.
>>
>> Of course, but
>> omega is somewhat ghostly.
>> Do the natnumbers reach till omega?
> 
> ω is an upper.bound of ℕᵈᵉᶠ.
> Of all upper.bounds of ℕᵈᵉᶠ, the lowest is ω.

Yes. Nevertheless almost all natural numbers are bewteen ℕᵈᵉᶠ and 
ω.
> 
> Each element of ℕᵈᵉᶠ is not upper.bound of ℕᵈᵉᶠ.
> No upper.bound of ℕᵈᵉᶠ is in ℕᵈᵉᶠ

Correct.
> 
>> Do the natnumbers reach till omega?
> 
> Define
> the natnumbers reach k  ⇔
> (∀ᵒʳᵈj≤k:(∃ᵒʳᵈi:j=i∪{i} ⇐ j≠0) ∧ 0<k) ∨ 0=k
> 
> k ∈ ω  ⇔  the natnumbers reach k
> The natnumbers only reach elements of ℕᵈᵉᶠ.

No.
> 
> ω, an upper.bound of ℕᵈᵉᶠ, is not.in ℕᵈᵉᶠ.
> The natnumbers do not reach ω

ω - 1 is the greates naturak number.
Easy to understand by the smallest unit fraction.

Regards, WM