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

On 19.10.2024 17:21, joes wrote:

Am Sat, 19 Oct 2024 15:38:53 +0200 schrieb WM:

On 19.10.2024 14:20, Alan Mackenzie wrote:

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

On 17.10.2024 23:22, Alan Mackenzie wrote:

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

 

When doubling natural numbers we obtain even numbers which have not
been doubled.
This is a sentence that every mathematician can understand.

It is not - it is ill formed and ambiguous.  It doesn't say which
natural numbers are being doubled.

That is not said because it is true for all sets of natural numbers.
Your critique is therefore not justified but due to your lack of
comprehension.

It is not true for infinite sets.

The density of numbers is halved when multiplying them by 2. That is true for all sets.

 

  It is unmathematical in that it
seems to posit a doubling being done one element at a time

Wrong again. Even all natural numbers can be multiplied by 2.

You seem unable to imagine that.

That's why I propose it?

 

rather than the standard mathematical concept of a mapping from N -> N
where n is mapped to 2n.  In this standard notion, all numbers are
doubled, and we encounter no undoubled even natural numbers.

Therefore the standard notion is wrong, if the natural numbers are a
set.

"if"

Yes.

 

The interval occupied by the numbers is doubled when all numbers are
multiplied by 2. If even the second half, which has not been multiplied,

You just said all numbers are multiplied. What is the "second half"?

When the density is halved and the number remains, then the interval is doubled.

Multiplying all n by 2 does not yield the same numbers.

In part it does. All even numbers are also naturals.

All even numbers which have been multiplied are natural numbers by definition. And by the same definition there were not more even natural numbers. But as the result there are more.
Regards, WM