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

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