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From: Moebius <invalid@example.invalid>
Newsgroups: sci.logic,sci.math
Subject: Re: Replacement of Cardinality
Date: Tue, 13 Aug 2024 19:51:19 +0200
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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.