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

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.

 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.

You cannot judge because you don't know that topic ....

I am a graduate in maths ....

Here is not discussed what you have studied. Remember, not even infinity
has been taught. Therefore you cannot judge.

Having studied maths, it is more likely that I am right.

.... and as fellow traveler can only parrot the words of matheologians
who are either too stupid to recognize or too dishonest to confess the
truth.

.... and able to understand and follow mathematical argument, and to
distinguish mathematical notions from pure hogwash.  [ Citation
restored after snipping. ]

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.

There is no connection between your second sentence and the intended
conclusion.  There are a lot of argument steps missing.

What I think your line of argument might be is that there "isn't room"
for an infinite number of >0 intervals to fit.  This simply isn't the
case.  Try adding up these intervals and you will find:
(1 - 1/2) = 1/2
(1 - 1/2) + (1/2 - 1/3) = 2/3
(1 - 1/2) + (1/2 - 1/3) + (1/3 - 1/4) = 3/4
(1 - 1/2) + (1/2 - 1/3) + (1/3 - 1/4) + (1/4 - 1/5) = 4/5
....
and so on.  Any finite sum of these is below 1.  The limit of the sum is
exactly 1.

In the same way, the infinite sum of a tail of that sequence can be made
small enough to be smaller than any x > 0, simply by starting at n = 1/x
(with appropriate rounding).  For example, if x = 0.01, we start the
sequence at
(1/100 - 1/101) + (1/101 - 1/102) + ......
This infinite sum sums to exactly 1/100, hence this infinitude of points
all fit in below x = 0.01.  The argument is the same for any x > 0.

If these terms had any significance, they would still be taught in
mathematics degree courses.

No, the teachers of such courses are too stupid or too dishonest.

Who do you think you are to accuse others of being stupid or dishonest?

I know that I have understood that topic better than the stupids.

Everybody else understands that the "stupids" are very bright indeed, and
collectively understand the topic far better than any non-specialist.

Otherwise, bright students would become aware of them and catch out
their teachers in inconsistencies.

They do. But every publishing is intercepted by the leading liars.

<Sigh> When I was an undergraduate, students published lots of
magazines, some of them about maths.  I'm sure they still do, though they
are likely to be online these days.  The "deceit" you think happens would
be exposed in these magazines, and thus become known,

You cannot believe that I am right, therefore you don't wish that I am
right, and you try to dismiss my argument.

Yet you cannot counter my argument, which is based on experience.

I know you are wrong.  Such overbearing censorship that you are picturing
just doesn't happen.  It couldn't happen.  Nobody rich enough and
powerful enough cares enough, or even at all.  If "dark numbers" acquired
military significance, they might, though.

Regards, WM

--
Alan Mackenzie (Nuremberg, Germany).