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From: WM <wolfgang.mueckenheim@tha.de>
Newsgroups: sci.math
Subject: Re: How many different unit fractions are lessorequal than all unit
 fractions?
Date: Sat, 5 Oct 2024 21:15:43 +0200
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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.

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".

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.

"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]

Regards, WM