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

WM <wolfgang.mueckenheim@tha.de> 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.

It is you who lacks a clear picture of infinity.

If all natnumbers are there and if 2n is greater than n, then the
doubled numbers do not fit into ℕ.

"Fitting into" a set is an entirely wrong way of conceiving of it.  Sets
have members.  They are not containers that may not be large enough.
For any finite n greater than zero, 2n is greater than n.  The same
does not hold for infinite n.

But note, that is only true if all natnumbers do exist.

Do you actually understand what it means for a mathematical entity to
exist?

All natural numbers exist by Peano's axioms.  Or more prosaically, you
cannot point to a natural number which doesn't exist.

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

No, you don't understand what existence means.  All natural numbers
exist, and have done since Peano, if not forever.  But that's drifting
off of mathematics into philosophy.

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.

No.  Not even close.

Regards, WM

--
Alan Mackenzie (Nuremberg, Germany).