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

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

On 05.10.2024 23:57, Alan Mackenzie wrote:

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

On 05.10.2024 15:57, Alan Mackenzie wrote:

Yes!  At least, sort of.  My understanding of "doesn't exist" is either
the concept is not (yet?) developed mathematically, or it leads to
contradictions.  WM's "dark numbers" certainly fall into the first
category, and possibly the second, too.

Definition: A natural number is "named" or "addressed" or "identified"
or "(individually) defined" or "instantiated" ....

That's five terms for the same thing.  Four of them (at least) are thus
redundant.  It is unmathematical to have such redundancy.

The vocabulary of my readers is very different. So the chance of
understanding is increased.

I doubt that.  If you cannot even teach them a coherent definition of a
single term, they are not the sort of readers who would understand
anyway.  Five (or six) terms for the same thing in mathematics can only
induce confusion and vagueness.

.... if it can be communicated, necessarily by a finite amount of
information, in the sense of Poincaré, such that sender and receiver
understand the same and can link it by a finite initial segment (1, 2,
3, ..., n) of natural numbers to the origin 0.

This is ridiculous!  It is so far removed from the austere simplicity of,
for example, Peano's axioms as to be thoroughly unmathematical.

Peano's axioms are invalid for large numbers.

Peano's axioms _define_ natural numbers.  They are thus valid for all
natural numbers by definition.  Any numbers for which they are invalid
are not natural numbers.

.... All other natural numbers are called dark natural numbers.  Dark
numbers are numbers that cannot be chosen as individuals.

Is "chosen" a sixth redundant word for "named", "addressed", ....?

Yes, all these words have the same meaning, but not all readers know all
words.

I think they're likely to know all of them in a normal English language
context if they know English at all.  Of course, they have different
meanings in English.  Making them have the same meaning in your variety
of mathematics can only lead to confusion.

"Chosen as individuals" isn't a mathemtical concept.  This phrase, as it
is written, makes it sound like the choice is being made by a conscious
individual person, according to something unspecified.  That doesn't
belong in mathematics.

It does not belong to the simple but inconsistent present mathematics.

It does not belong in any mathematics.  The insinuation that present
mathematics is inconsistent is unwarranted.  It's not.

Communication can occur
- by direct description in the unary system like ||||||| or as many
beeps, raps, or flashes,
- by a finite initial segment of natural numbers (1, 2, 3, 4, 5, 6, 7),
- as n-ary representation, for instance binary 111 or decimal 7,
- by indirect description like "the number of colours of the rainbow",
- by other words known to sender and receiver like "seven".

The existence of natural numbers is independent of their communication by
people.

You are a believer in God having created them?

You're twisting my meaning here.  All natural numbers exist, by Peano's
axioms.  Any particular natural number exists independently of whether it
has been, or even could be, communicated between humans.

Only when a number n is identified we can use it in mathematical
discourse ....

This is something you haven't proved.

Try to use another number.

That "answer" bears no relationship to my point.  You haven't proved
anything about "identified" numbers, you can't, and you're not going to.

Besides, mathematicians routinely use "unidentified" numbers in
discourse.  For example "If p is a prime number of the form 4m + 1, it is
the sum of two squares.".  That is a statement about an infinite number
of numbers, none of which are "identified".

It is not a statement about an individual number but about a set, some
of which can be defined.

Don't be so silly!  It's a statement about each of the numbers in an
infinite set.  To repeat, it's a statement about "unidentified" numbers,
made by mathematicians, hence is a counter example to your assertion that
only "identified" numbers can be used in mathematical discourse.

.... and can determine the trichotomy properties of n and of every
multiple k*n or power n^k or power tower k_^n with respect to every
identified number k. ℕdef contains all defined natural numbers as
elements – and nothing else. ℕdef is a potentially infinite set;
therefore henceforth it will be called a collection.

All natural numbers are "defined" in your sense of that word.

No, after every identified number almost all are following, almost all
of them are dark.

As I said, all natural number are defined, as I have proved below.

As a proof, we only need note that every non-empty subset of N has a
least member.  Suppose there is a non-empty set of "undefined" natural
numbers.  Then there is a least such number.

No. The identified numbers are a potentially infinite collection.
Therefore there is no least dark number.

The "unidentified" natural numbers are a subset of N.  Otherwise this
whole discussion is meaningless.

You're confusing yourself with the "potential" in potential infinity,
which shows why the outmoded term is such a bad idea.  The set of
"identified" numbers is either finite or infinite.  Given that N is
infinite, if the set of "identified" numbers is finite, its complement
must exist and be non-empty.  It therefore has a least member, which is
thus "identified".  Contradiction.

I first came across the terms "potential infinity" and "actual
infinity" on this newsgroup, not in my degree course a few decades
ago.

It is carefully avoided because closer inspection shows contradictions.

There are no such contradictions.

There are many. For instance: 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.  Hint: Skilled mathematicians have worked on trying to
prove the inconsistency of maths, without success.  You (not a
mathematician) or I don't stand a chance of succeeding in this.

Therefore set theorists use just what they can defend. If actual
infinity is shown self contradictory (without dark numbers), then they
evade to potential infinity temporarily which has no completed sets and
cannot complete bijections.

Not really.  There is simply no need for "actual" and "potential"
infinity.

You have not learnt about it during study and obviously not afterwards.

I've learnt about it here on sci.math.  So far, nobody's produced a
situation where the difference between "actual" and "potential" infinity
makes a difference.  Historically, the terms have fallen into disuse,
because they're not useful in maths.

They are relics from the past, from before the time when
mathematicians understood infinity as they do today.

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

I am a graduate in maths ....

.... 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.

"You use terms like completed versus potential infinity, which are
not part of the modern vernacular." [P.L. Clark in "Physicists can be
wrong", tea.MathOverflow (2 Jul 2010)] This is the typical reproach
to be expected when the different kinds of infinity are analyzed and
taught.

Here the difference is clearly stated:
"Should we briefly characterize the new view of the infinite introduced
by Cantor, we could certainly say: In analysis we have to deal only with
the infinitely small and the infinitely large as a limit-notion, as
something becoming, emerging, produced, i.e., as we put it, with the
potential infinite. But this is not the proper infinite. That we have
for instance when we consider the entirety of the numbers 1, 2, 3, 4,
... itself as a completed unit, or the points of a line as an entirety
of things which is completely available. That sort of infinity is named
actual infinite." [D. Hilbert: "Über das Unendliche", Mathematische
Annalen 95 (1925) p. 167]

That's from 1925.  It is not a modern understanding of the infinite.

But a correct one. The modern understanding is pure deceit.

No, the modern one is a result of research and improved understanding.
Deceit would not survive long before being unmasked.

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?

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, both to other
students and the deceiver's colleagues, and eventually to the public at
large.  There is no system of censorship in place which could counter
this.  Honest academics (the vast majority) would not risk it.

Regards, WM

--
Alan Mackenzie (Nuremberg, Germany).