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

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

On 06.10.2024 18:48, Alan Mackenzie wrote:

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

On 06.10.2024 15:59, Alan Mackenzie wrote:

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

All unit fractions are separate points on
the positive real axis, but there are infinitely many for every x > 0.
That can only hold for definable x, not for all.

Poppycock!  You'll have to do better than that to provide such a
contradiction.

It is good enough, but you can't understand.

I do understand.  I understand that what you are writing is not maths.
I'm trying to explain to you why.  I've already proved that there are no
"undefinable" natural numbers.  So assertions about them can not make any
sense.

You have not understood that all unit fractions are separate points on
the positive axis.

I understand that full well.  I have a reasonable grasp of point set
topology.  You don't.

Every point is a singleton set and could be seen as such, but it
cannot. Hence it is dark.

That's a complete non-sequitur.  In fact, it's gobbledegook.  Points
aren't sets.  What "it cannot" refers to is more than unclear.  The same
applies to the "it" in "it is dark".

"Dark" would appear to be a seventh synonym for (the negative of)
"defined" ....

  Hint: Skilled mathematicians have worked on trying to
prove the inconsistency of maths, without success.

What shall that prove? Try to understand.

It shows that any such results are vanishingly unlikely to be found by
non-specialists such as you and I.

Unlikely is not impossible.

As near impossible as can be without actually being there.  You cannot
prove the inconsistency of maths: your understanding of the basics is far
too limited.  You don't understand the infinite; you don't understand
point set topology; you don't understand basic set theory.  In fact, your
understanding is so limited, that you have no idea of the extent of your
ignorance.  If you had graduated in mathematics you would have a better
idea of all these things.  But you didn't.

Try only to understand my argument. ∀n ∈ ℕ: 1/n - 1/(n+1) > 0. How can
infinitely many unit fractions appear before every x > 0?

You are getting confused with quantifiers, here.  For each such x, there
is an infinite set of fractions less than x.  For different x's that set
varies.  There is no such infinite set which appears before every x > 0.

The set varies but infinitely many elements remain the same.

That is not true.  There is no element which is in every one of these
sets.

A shrinking infinite set which remains infinite has an infinite core.

Again, no.  There is no such thing as a "core", here.  Each of these sets
has an infinitude of elements.  No element is in all of these sets.

Regards, WM

--
Alan Mackenzie (Nuremberg, Germany).