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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? (infinitary)
Date: Sun, 6 Oct 2024 15:52:29 +0200
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On 06.10.2024 12:16, Alan Mackenzie wrote:

> 
> You mean, that there is a difference?  I remain unconvinced.

"Numerals constitute a potential infinity. Given any numeral, we can 
construct a new numeral by prefixing it with S. Now imagine this 
potential infinity to be completed. Imagine the inexhaustible process of 
constructing numerals somehow to have been finished, and call the result 
the set of all numbers, denoted by . Thus  is thought to be an actual 
infinity or a completed infinity. This is curious terminology, since the 
etymology of 'infinite' is 'not finished'." [E. Nelson: "Hilbert's 
mistake" (2007) p. 3]

"A potential infinity is a quantity which is finite but indefinitely 
large. For instance, when we enumerate the natural numbers as 0, 1, 2, 
...., n, n+1, ..., the enumeration is finite at any point in time, but it 
grows indefinitely and without bound. [...] An actual infinity is a 
completed infinite totality. Examples: , , C[0, 1], L2[0, 1], etc. 
Other examples: gods, devils, etc." [S.G. Simpson: "Potential versus 
actual infinity: Insights from reverse mathematics" (2015)]

"Potential infinity refers to a procedure that gets closer and closer 
to, but never quite reaches, an infinite end. For instance, the sequence 
of numbers 1, 2, 3, 4, ... gets higher and higher, but it has no end; it 
never gets to infinity. Infinity is just an indication of a direction – 
it's 'somewhere off in the distance'. Chasing this kind of infinity is 
like chasing a rainbow or trying to sail to the edge of the world – you 
may think you see it in the distance, but when you get to where you 
thought it was, you see it is still further away. Geometrically, imagine 
an infinitely long straight line; then 'infinity' is off at the 'end' of 
the line. Analogous procedures are given by limits in calculus, whether 
they use infinity or not. For example, limx0(sinx)/x = 1. This means 
that when we choose values of x that are closer and closer to zero, but 
never quite equal to zero, then (sinx)/x gets closer and closer to one.
	Completed infinity, or actual infinity, is an infinity that one 
actually reaches; the process is already done. For instance, let's put 
braces around that sequence mentioned earlier: {1, 2, 3, 4, ...}. With 
this notation, we are indicating the set of all positive integers. This 
is just one object, a set. But that set has infinitely many members. By 
that I don't mean that it has a large finite number of members and it 
keeps getting more members. Rather, I mean that it already has 
infinitely many members.
	We can also indicate the completed infinity geometrically. For 
instance, the diagram at right shows a one-to-one correspondence between 
points on an infinitely long line and points on a semicircle. There are 
no points for plus or minus infinity on the line, but it is natural to 
attach those 'numbers' to the endpoints of the semicircle.
	Isn't that 'cheating', to simply add numbers in this fashion? Not 
really; it just depends on what we want to use those numbers for. For 
instance, f(x) = 1/(1 + x2) is a continuous function defined for all 
real numbers x, and it also tends to a limit of 0 when x 'goes to' plus 
or minus infinity (in the sense of potential infinity, described 
earlier). Consequently, if we add those two 'numbers' to the real line, 
to get the so-called 'extended real line', and we equip that set with 
the same topology as that of the closed semicircle (i.e., the semicircle 
including the endpoints), then the function f is continuous everywhere 
on the extended real line." [E. Schechter: "Potential versus completed 
infinity: Its history and controversy" (5 Dec 2009)]

Regards, WM