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

On 20.10.2024 23:41, Jim Burns wrote:

On 10/20/2024 4:42 PM, WM wrote:

On 20.10.2024 22:23, Jim Burns wrote:

 

If n is in the set ℕ≠{1,...,n} of finites,
then 2⋅n is in the set ℕ≠{1,...,n} of finites.

>
When the elements of any set of naturals are multiplied by 2,

 ∃{1,2,...,n-1,n}  ⇔
∃{1,2,...,n-1,n,n+1,...,n+n-1,n+n}
 

their density is halved,
their reality remains the same,
their extension is doubled.
Therefore
the image contains numbers which are not in the range.

 n ∈ ℕ  ⇔  ∃{1,2,...,n-1,n}

n is not among the dark numbers.

 ∃{1,2,...,n-1,n}  ⇔  ∃{1,2,...,n-1,n,n+1,...,n+n-1,n+1}
 ∃{1,2,...,n-1,n,n+1,...,n+n-1,n+1}  ⇔  n+n ∈ ℕ
 n ∈ ℕ  ⇔  n+n ∈ ℕ

That and more holds for all definable numbers. But: When doubling numbers, their distance increases in positive direction, hence larger numbers are in the image than in the range. If the range was complete, the image shows that the range was not complete.
Regards, WM