Re: The reality of sets, on a scale of 1 to 10 [Was: The non-existence of "dark numbers"]

On 22.03.2025 16:28, Alan Mackenzie wrote:

The term "potentially infinite" has no use in
mathematics.  Philosophers, etc., might cling to it, though.

For your education.:
"On account of the matter I would like to add that in conventional mathematics, in particular in differential- and integral calculus, you can gain little or no information about the transfinite because here the potential infinite plays the important role, I don't say the only role but the role emerging to surface (which most mathematicians are readily satisfied with). Even Leibniz with whom I don't harmonize in many other respects too, has [...] fallen into most spectacular contradictions with respect to the actual infinite." [G. Cantor, letter to A. Schmid (26 Mar 1887)]
"Nevertheless the transfinite cannot be considered a subsection of what is usually called 'potentially infinite'. Because the latter is not (like every individual transfinite and in general everything due to an 'idea divina') determined in itself, fixed, and unchangeable, but a finite in the process of change, having in each of its current states a finite size; like, for instance, the temporal duration since the beginning of the world, which, when measured in some time-unit, for instance a year, is finite in every moment, but always growing beyond all finite limits, without ever becoming really infinitely large." [G. Cantor, letter to I. Jeiler (13 Oct 1895)]
"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]
"We introduce numbers for counting. This does not at all imply the infinity of numbers. For, in what way should we ever arrive at infinitely-many countable things? [...] In philosophical terminology we say that the infinite of the number sequence is only potential, i.e., existing only as a possibility." [P. Lorenzen: "Das Aktual-Unendliche in der Mathematik", Philosophia naturalis 4 (1957) p. 4f]
"Until then, no one envisioned the possibility that infinities come in different sizes, and moreover, mathematicians had no use for 'actual infinity'. The arguments using infinity, including the Differential Calculus of Newton and Leibniz, do not require the use of infinite sets." [T. Jech: "Set theory", Stanford Encyclopedia of Philosophy (2002)]
"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." [E. Schechter: "Potential versus completed infinity: Its history and controversy" (5 Dec 2009)]
Will you show gratitude to be educated in great detail?
Regards, WM