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

On 09.10.2024 21:13, Alan Mackenzie wrote:

WM <invalid@no.org> wrote:

Am 09.10.2024 um 18:12 schrieb Alan Mackenzie:

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

 

You've misunderstood the nature of N.  The set is not
{1, 2, 3, ..., ω}, it is {1, 2, 3, ...}.

 

I use ℕ U {ω} for clarity.

 You would do better not to do so.  It gives wrong results.

It gives unfamiliar results because you have no clear picture of actual infinity. If all natnumbers are there and if 2n is greater than n, then the doubled numbers do not fit into ℕ. But note, that is only true if all natnumbers do exist.
If not all do exist, te doubling yields larger natnumbers, some of which have not existed before. But that means potential infinity.

Numbers multiplied by 2 do not remain unchanged. That is not intuition
but mathematics.

 True,

Fine, then you can follow the above discussion. Either doubling creates new natural numbers. Then not all have been doubled. Or all have been doubled, then some products fall outside of ℕ.
Regards, WM