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

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

On 19.10.2024 20:22, Alan Mackenzie wrote:

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

An infinite set is one which has a proper subset which can be
put into 1-1 correspondence with the original set.  That is the
definition.

According to Dedekind every set {1, 2, 3, ..., n} is in correspondence
with the set {2, 4, 6, ..., 2n} which covers twice the interval,
containing numbers not in the original set.

This is true, though has nothing to do with my point about the
definition of an infinite set.

This does not change when the whole set ℕ is multiplied by 2.

It does.  It changes dramatically.

The result covers twice the interval, ....

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

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

Regards, WM

--
Alan Mackenzie (Nuremberg, Germany).