Re: Replacement of Cardinality

Le 21/08/2024 à 20:20, Jim Burns a écrit :

On 8/21/2024 6:43 AM, WM wrote:

Le 20/08/2024 à 22:05, Jim Burns a écrit :

 

No min.⅟ℕᵈᵉᶠ exists.

>
The reason is potential infinity.

 Is potentialᵂᴹ.infinity mathematicsᵂᴹ?

All of classical mathematics is embedded into potential infinity.

 Judging from your edicts on this topic,
the answer looks to be both 'yes' and 'no'.

I am interested in actual infinity. We cannot prove its existence but only apply logic like this: If a complete chain exists which does not continue into the negative, then it ends before. Either at one or at more members. The latter is excluded by ∀n ∈ ℕ: 1/n - 1/(n+1) > 0 .

  We are agreed, though, that
no min.⅟ℕᵈᵉᶠ exists.

Of course. After each one there will appear more. Choose the smallest one you can, recognize that there are many more following.

 It follows that ⅟ℕᵈᵉᶠ has properties that
the unit.fractionsᵈᵉᶠ > x > 0
don't have.

There is no set ℕᵈᵉᶠ because of lacking completeness. Sets of set theory are complete.
Regards, WM