Re: Replacement of Cardinality

Am 13.08.2024 um 19:02 schrieb Jim Burns:

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

either in a step of size 1
or in a step of size more than 1.

Let's "assume" that this is true (sort of) for the sake of the argument.

But increase by more than 1 is excluded by
the gaps between unit fractions.

I can't see any argument for this claim.
Actually, for each and every real number x > 0 there are infinitely many unit fractions smaller than x: 1/ceil(1/x + 1)), 1/ceil(1/x + 2)), 1/ceil(1/x + 3)), ...
Hence the difference between NUF(0) (i.e. 0) and NUF(x) is "infinite" for each and every x e IR, x > 0.
In fact, Ax > 0: NUF(x) = aleph_0, while NUF(0) = 0.