Re: How many different unit fractions are lessorequal than all unit fractions?

On 9/5/24 10:03 AM, WM wrote:

Le 05/09/2024 à 02:35, Richard Damon a écrit :

On 9/4/24 3:10 PM, WM wrote:

 

Either ∀n ∈ ℕ: 1/n - 1/(n+1) > 0 is wrong or Peano is wrong.

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Why?
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Peano has been generalized from the small natural numbers.
"All different unit fractions are different" however is a basic truth. Therefore I accept the latter.

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And all ARE different as there is always a space between them, but that space gets arbitrary small (but still finite.)

 NUF(x) increases from 0 to ℵo. But it cannot increase to ℵo at one point x > 0 because at every point there is at most one unit fraction. .Before ℵo there come 1, 2, 3, ...
 Regards, WM
 

But since there is not "first" point, NUF(x) just doesn't exist if the domain is just the finite values.
NUF(x) can only be defined if x is in some post-finite set that defines values as unit fractions in that post-finite value.
Things like 1/(w-1), 1/(w-2), ...
This mean that when we "count" we count as:
0, 1, 2, 3, 4, .... w-4, w-3, w-2, w-1, w, w+1, w+2,...
where we shift from the natural numbers to your post-finite values as we pass through the ... and then into that "normal" transfinite value when we get to w.