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

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

On 19.10.2024 22:19, Alan Mackenzie wrote:

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

The result covers twice the interval, ....

It does not, except in the sense that twice infinite = infinite.

Twice the infinite contains numbers not in the first infinite.

It does not.

What else could twice mean?

It doesn't really mean much at all.  I was trying to correct your
intuition.

.... containing numbers not in the original set ℕ.

No.  If you think that, then give an example of a 2n which "isn't in the
original set N".

These numbers are dark. Their existence is proven by the fact that
∀n ∈ ℕ, n < 2n.

Their non-existence I proved in this thread, many posts ago.

You won't and you can't.  But you'll likely come back
to your standard get-out clause about (non existent) "dark numbers".

The only alternative is that doubling numbers creates only doubled
numbers.

That's more or less what happens, except that numbers aren't "created".
They just are.

It is not acceptable.

It is accepted by mathematicians.

It cannot be apologized by the possibility to map all numbers on even
numbers.

There exists a 1-1 correspondence between all natural numbers and all
even natural numbers, a proper subset of all natural numbers.  The set of
all natural numbers is thus infinite.

Not all properties of finite sets hold for infinite sets.

Regards, WM

--
Alan Mackenzie (Nuremberg, Germany).