Re: Replacement of Cardinality

Le 02/08/2024 à 17:17, Richard Damon a écrit :

On 8/2/24 10:58 AM, WM wrote:

It is not a contradiction to my formula if some n has no n+1.

>
It is a violation of the DEFINITION of the Natural Numbers.

 Otherwise there is a contradiction of mathematics: separated unit fractions.

 No, all Unit fractions are separated in distance by a finite amount, it is just that you can't specify any finite distance that separates all unit fractions.

I don't try. But I know that one unit fraction must be the first.

 The reversal of the order of qualifications causes the problem.

Distances on the real line imply an order, even it it cannot be discerned.

 Every number 1/n is separated from the next smaller unit fraction, 1/(n+1) by a distance of 1/(n*(n+1)) which is a value that is greater than zero, so we always have a finite difference between all unit fractions, but that distance gets arbitrarily small,

Can they get smaller than 2^ℵ₀ points?
If not ℵ₀ unit fractions need ℵ₀*2^ℵ₀ points above zero. For those x > 0 the claim
∀x > 0:  NUF(x) ≥ ℵ₀
is wrong.
Regards, WM