Re: Replacement of Cardinality

On 8/19/24 7:54 AM, WM wrote:

Le 18/08/2024 à 10:16, joes a écrit :

Am Sat, 17 Aug 2024 13:42:03 +0000 schrieb WM:

 

Try to choose an eps smaller infinitely many unit fractions. Fail.

Why should it be possible?

 Because when infinitely many exist, then also finitely many must exist - according to logic and mathematics: ∀n ∈ ℕ: 1/n - 1/(n+1) > 0 . Everything else is the superstition of matheologians.
 Regards, WM

But why should the finite sub-set need to include the (non-existant) first of the infinite set.
Yes, your finite "natnumbers" which have a highest vaule, will have a lowest unit fraction whose 1/n doesn't have a 1/(n+1) with in it,
But the FULL set of unit fractions based on the INFINITE set of Natural Numbers WILL include that n+1, and thus that 1/(n+1) so the 1/n wasn't the first unit fraction of the full set, only your finite sub-set that no one else cares about.
When you ADMIT that you are talking about a finite set, and thus something that CAN'T be the actual set of Natural Numbers, why do we care.
And if NUF(x) counts the unit fractions of NatNumbers, then NUF(1) won't be aleph_0, but the size of that set of NatNumbers.