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From: WM <wolfgang.mueckenheim@tha.de>
Newsgroups: sci.math
Subject: Re: The reality of sets, on a scale of 1 to 10 [Was: The
 non-existence of "dark numbers"]
Date: Sun, 23 Mar 2025 20:22:27 +0100
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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