Re: Replacement of Cardinality

On 8/21/24 8:32 AM, WM wrote:

Le 21/08/2024 à 13:32, Richard Damon a écrit :

On 8/21/24 6:44 AM, WM wrote:

Le 20/08/2024 à 23:25, FromTheRafters a écrit :

WM explained :

Le 20/08/2024 à 12:31, FromTheRafters a écrit :

on 8/19/2024, Richard Damon supposed :

>

You can not derive a first number > 0 in any of the Number System that we have been talking about, Unit Fractions, Rationals or Reals, so you can't claim it to exist.

>
Not in their natural ordering.

>
Dark numbers have no discernible order. It is impossible to find the smallest unit fraction or the next one or the next one. It is only possible to prove that NUF(x) grows by 1 at every unit fraction. It starts from 0.

>

Normally, the unit fractions are listed in the sequence one over one, one over two, one over three etcetera. There is a first but no last. Now you have started from the wrong 'end'

>
No, I have started from the other end. It exists at x > 0 because NUF(0) = 0.

 

But the other end doesn't "begin" with a first Natural Number Unit fraction, if it has a beginning that will be a trans-finite number.

 No, it is a finite number. ∀n ∈ ℕ: 1/n - 1/(n+1) > 0 holds for all and only reciprocals of natural numbers.
 Regards, WM
 

Can't be, because if it WAS 1/n, then 1/(n+1) would be before it, and thus your claim is wrong. If 1/(n+1) wasn't smaller than 1/n, then we just have that 1/n - 1/(n+1) wouldn't be > 0, so it can't be.
BY DEFINITION, there is no "Highest" Natural Number, if n exists, so does n+1, and your formula says you accept that n+1 exists, or you couldn't use it.
If you don't have that property, you don't have the Natural Numbers.
PERIOD.
DEFINITION.
If you claim your mathematics say it can't be, then your mathematics were just proven to not be abble to handle the unbounded set of the Natural Numbers.
Sorry, that is just the facts.