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

On 17.10.2024 00:39, Jim Burns wrote:

No natural number is
the first to not.have a natural.number.double.

True.

>

When doubling all natural numbers
we obtain only natural numbers.

>
That is impossible.

 There is no first natural number from which we obtain
(by doubling) anything not.a.natural.number.

True.

 The only set of natural numbers with no first
is the empty set..

No, the set of dark numbers is another set without smallest element.

 There is no ▒▒▒▒▒ natural number from which we obtain
(by doubling) anything not.a.natural.number.

Correct is: There is no such _definable_ natnumber.
There is a general rule not open to further discussion:
When doubling natural numbers we obtain even numbers which have not been doubled.
In potential infinity we obtain more even natural numbers than have been doubled.
In actual infinity we double ℕ and obtain neither ℕ or a subset of ℕ.
Regards WM