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

On 01.11.2024 13:33, FromTheRafters wrote:

After serious thinking WM wrote :

On 01.11.2024 11:57, FromTheRafters wrote:

It happens that WM formulated :

On 01.11.2024 00:43, Richard Damon wrote:

On 10/31/24 1:35 PM, WM wrote:

On 31.10.2024 12:36, Richard Damon wrote:

On 10/30/24 11:38 AM, WM wrote:

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NUF(x) MUST jump from 0 to Aleph_0 at all real values x, as below ANY real number x, there are Aleph_0 unit fractions.

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You cannot distinguish them by any real number? That proves that they are dark.

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They are not finite values.

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All unit fractions are finite values.

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Each unit fraction is finite, the set of all unit fractions is not finite. Not finite is 'infinite' and there is no potential or actual anymore -- just finite and not finite.

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Actual means that all are there,

 They are all there anyway, by definition.
 

including the smallest.

 No smallest, since you simply inverted the set of naturals which has no largest.

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If an invariable set of numbers is there, then there is a smallest and a largest number of those which are existing.

Infinite subsets don't do that for you, even if you wish really hard.

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They cannot evade if they are invariable.

It is a point on the real line, well separated from its neighbour.

 No point on the real line is separated, that is for discrete sets.

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All unit fractions belong to points which are separated by non-unit fractions.