Re: Replacement of Cardinality

Am 05.08.2024 um 01:07 schrieb Chris M. Thomasson:

On 8/4/2024 8:35 AM, WM wrote:

Le 04/08/2024 à 02:15, Moebius a écrit :

Am 03.08.2024 um 21:54 schrieb Jim Burns:

On 8/3/2024 10:23 AM, WM wrote:

>

NUF(x) = ℵ₀ for all x > 0 is wrong.

>
Nonsense.
>
Actually, Ax > 0: NUF(x) = ℵ₀.

>
You mean that there are ℵ₀ unit fractions smaller than all positive x?

Obviously not.
What I mean is that for all positive x there are ℵ₀ unit fractions smaller than x.

Impossible. [...] Not even one unit fraction can be smaller than all positive x.

No one (except WM) claimed that there's a unit fraction which is smaller than all positive x.

Huh?

WM is constantly mixing up
          ∀x > 0: ∃^ℵ₀ u ∈ ⅟ℕ: u < x   (true)
with
          ∃^ℵ₀ u ∈ ⅟ℕ: ∀x > 0: u < x   (false) .
(Here ⅟ℕ = {1/n : n e IN} is the set of all unit fractions.)

Say x = 1/2, there are infinite smaller unit fractions, say, 1/4, 1/5, 1/6, ect... However there is only one larger one, 1/1. See? No smallest one for 1/0 is not a unit fraction! There is a largest one, 1/1...
 They tend to zero, but there is no smallest one...

Yeah.
Proof: If s is a unit fraction then 1/(1/s + 1) is a unit fraction which is smaller than s (for each and every s).

See?