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

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.

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.

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.

 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. If all natural numbers have been mapped, then there are larger numbers in the image. That is an unavoidable consequence. These numbers cannot be seen, but that does not negate the consequence.
If this is not accepted, then not all natural numbers of the image have been in the original set.
Regards, WM