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

WM <wolfgang.mueckenheim@tha.de> wrote:

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.

No.  It is the empty set (which I proved many posts ago).  All empty sets
are the same - there is only one of them.

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

Correct is: There is no such _definable_ natnumber.

That is the only sort of natural number there is.

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.

All of these "rules" are so loosely and ambiguously formulated, that they
don't actually say anything at all - they are meaningless.

Regards WM

--
Alan Mackenzie (Nuremberg, Germany).