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

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

On 20.10.2024 11:21, Alan Mackenzie wrote:

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

On 20.10.2024 00:54, Jim Burns wrote:

On 10/19/2024 2:19 PM, WM wrote:

A doubled finite is finite.

If all finites are doubled, then not all results can be in that set.
Either more finites appear, or the results are infinite.

That's your intuition getting the better of you again.

It is not intuition but fact that 2n > n.

That's not where you're going wrong.  You're confusing the doubling of a
single number n, with the mapping of an infinite set N into the same set
N.

When "all finites" (by which I assume you mean natural numbers) are
doubled, all the doubled numbers are finite, too.

Maybe, but they are not all in the original set. Hence more natural
numbers are necessary than have been doubled or mapped.

You're confusing yourself by using intuition from finite sets and
misapplying it to infinite sets.  You're simply wrong, there.

We're talking about a mapping between infinite sets, not a process.
Nothing "appears".

Say mapping or multiplying, it is a process. But that is not important.
Important is only that in the image there are numbers which have not
been mapped because 2n > n.

That "because" attempts to connect two unrelated statements there are no
"numbers which have not been mapped".  If you think there are, give an
example.

If you think some of the doubled numbers are infinite, please give an
example of a natural number which when doubled becomes infinite.

In the image there are numbers which are not in the original set.

There are not.  If you think there are, give an example.

If all natural numbers have been mapped, then there are larger numbers
in the image. That is an unavoidable consequence.

Larger than what?  There are larger numbers in the image than any number
in the image.  There are larger numbers in the image than any number in
the range.  So what?

These numbers cannot be seen, but that does not negate the consequence.

These (dark) numbers don't exist, and there are no consequences to
negate.

If this is not accepted, then not all natural numbers of the image have
been in the original set.

<Sigh>.  The sets are infinite.  Your intuition is not.

Regards, WM

--
Alan Mackenzie (Nuremberg, Germany).