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

Am Sat, 05 Oct 2024 21:15:43 +0200 schrieb WM:

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

That is possible for all natural numbers.

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".

Where did you get this idea from?

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

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.
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.

Seems sensible not to use the contradictory distinction between
potential and actual.

"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.

They are not taught anymore.

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]

--
Am Sat, 20 Jul 2024 12:35:31 +0000 schrieb WM in sci.math:
It is not guaranteed that n+1 exists for every n.