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

On 9/7/24 9:08 AM, WM wrote:

On 07.09.2024 04:01, Richard Damon wrote:

On 9/6/24 4:40 PM, WM wrote:

On 06.09.2024 14:38, joes wrote:

Am Fri, 06 Sep 2024 14:12:29 +0200 schrieb WM:

>

0 < ... < ⅟⌊4+⅟x⌋ < ⅟⌊3+⅟x⌋ < ⅟⌊2+⅟x⌋ < ⅟⌊1+⅟x⌋ < x

0 is smaller than all that. Therefore there is no increase at 0.
x is larger than all that. Therefore your x is not the least one
posiible.

Therefore no x can be the least unit fraction.

>
No definable x. No epsilon.

>
Which means no "Unit Fraction" as the reciprical of a Natural Number, since they are all definable.

 All natural numbers which you recognize are definable.

Which means that *ALL* Natural numbers are definable, the whole infinite set of them.

>
Thus, your NUF must also be counting some sub-finite values, or it just doesn't exist.

 It exists. It counts definable numbers as well as sub-finitely definale numbers.

And thus you accept that the domain of our NUF(x) isn't just a finite number system, but includes a sub-finite system of numbers, and that the value of x where NUF(x) is 1 isn't in the domain of finite numbers, so it doesn't increase at 1/n for some Natural Number, but increase at some "unit fraction" that is the reciprical of an post-finite number, greater than all Natural numbers.

 Regards, WM