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

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.

Should all places ω+2, ω+4, ω+6, ... remain empty?

It's not clear what you mean by this.  There are no such "places".

According to Cantor they are there in actual infinity.

More like according to your misunderstanding of Cantor.  There are no
such things as "places" in sets, just elements.  There certainly aren't
"places" in a set that, somehow, can be "occupied" or "empty".

Should the even numbers in spite of doubling remain below ω?

Yes, of course.

Then they must occupy places not existing before.

No.  Remember the set is infinite, so you cannot use finite intuition to
reason about it.

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

True, but your paragraph has absolutely nothing to do with my paragraph
it puportedly answers.

That means the original set had not contained all natural numbers. That
mans no actual or complete infinity.

Nonsense.

Sorry, you are a real believer but cannot discuss rational arguments.

It has nothing to do with "belief".  It so happens I studied this
elementary stuff under the guidance of those who understood it, indeed
had possibly researched it, at a time when my brain was flexible enough
to understand and absorb it, which I did.

You have had none of these advantages, and your efforts later in life to
pick up set theory have not borne fruit.  Let's face it, set theory, even
at the elementary level, is not your thing.  You don't understand it, you
can't understand it, and you are no longer capable of learning it.  Why
can't you accept that?  Day to day living doesn't require set theory.

Regards, WM

--
Alan Mackenzie (Nuremberg, Germany).