Re: Replacement of Cardinality

Am Wed, 14 Aug 2024 12:26:27 +0000 schrieb WM:

Le 13/08/2024 à 19:32, Moebius a écrit :

Am 13.08.2024 um 19:02 schrieb Jim Burns:

On 8/13/2024 10:21 AM, WM wrote:

Le 12/08/2024 à 19:44, Jim Burns a écrit :

 

There is no ⅟nₓ before the end of the positive axis without ⅟(nₓ+1)
before the end of the positive axis.

You cannot see it. It is dark.

No number can be seen, dark or not.

Numbers can be seen.

What is "seeing"?

resulting in a real coordinate x with NUF(x) = 1.

Assume NUF(x0) = 1 with x0 e IR. This means that there is exactly one
unit fraction u such that u < x0. Let's call this unit fraction u0.
Then (by definition) there is a (actually exactly one) natural number n
such that u0 = 1/n. Let n0 e IN such that u0 = 1/n0. But then (again by
definition) 1/(n0 + 1) is a unit fraction which is smaller than u0 and
hence smaller than x0. Hence NUF(x0) > 1. Contradiction!

Therefore 1/(n0 + 1) does not exist.

That's one way to resolve it. But how do you do mathematics without that
axiom?

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Am Sat, 20 Jul 2024 12:35:31 +0000 schrieb WM in sci.math:
It is not guaranteed that n+1 exists for every n.