Re: Replacement of Cardinality

On 8/4/2024 5:24 PM, Moebius wrote:

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).

Yup.
always_smaller(i) = (1/(i + 1))
So, starting at 1/1:
always_smaller(1) = 1/2
always_smaller(2) = 1/3
always_smaller(3) = 1/4
always_smaller(4) = 1/5
...
;^)

 

See?