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

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.

.... 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.  Such a
definition might have its place in sociology or even philosophy, but not
mathematics.

.... 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", ....?

"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.

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.  Adopting your notions into number theory would make that theory
hopelessly awkward and cumbersome and barely, if at all, capable of
discovering all the fascinating things about numbers that it has done.

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

This is something you haven't proved.  Given how woolly your definition
of "identified" is, it's probably something incapable of proof.

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".

.... 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.  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.  The fact of being this least number
is its definition.  We thus have a natural number which is both undefined
and defined.  This is a contradiction.  Therefore the assumption of a
non-empty set of "undefined" numbers must be false.

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.

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

"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.

If these terms had any significance, they would still be taught in
mathematics degree courses.  Otherwise, bright students would become
aware of them and catch out their teachers in inconsistencies.  Some such
students are almost incredibly bright, and catching out teachers is
something in the nature of a sport.  It happens rarely, but is satisfying
for all concerned when it does happen.

Regards, WM

--
Alan Mackenzie (Nuremberg, Germany).