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

On 10/10/24 2:32 PM, WM wrote:

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.

No, *YOU* don't understand what actual infinity is like as you picture it just like the finite, but bigger.

 If not all do exist, te doubling yields larger natnumbers, some of which have not existed before. But that means potential infinity.

But there aren't any natural numbers that have not existed before. That is just showing that your "actual infinity" is a finite set that you grabbed on the way to infinity, but didn't get there yet.
Actual infinity means you need to wait until you get there. The problem is finite creatures can't do that, so can't handle actual 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
 

You may think that, but it isn't true. That is just your actually nearly infinity that you think is actual infinity.