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

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