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From: joes <noreply@example.org>
Newsgroups: sci.math
Subject: Re: How many different unit fractions are lessorequal than all unit
 fractions?
Date: Sun, 6 Oct 2024 11:28:36 -0000 (UTC)
Organization: i2pn2 (i2pn.org)
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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.