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From: Richard Damon <richard@damon-family.org>
Newsgroups: sci.math
Subject: Re: How many different unit fractions are lessorequal than all unit
 fractions? (infinitary)
Date: Sun, 6 Oct 2024 13:19:18 -0400
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On 10/6/24 9:52 AM, WM wrote:
> 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
> 

And the problem with your "Actual Infinity" is since it requires that an 
infinite amount of "work" was done to create it, only an actually 
infinite being can experience it.

Such a being would understand the nature of such a set, and not presume 
it acts like the finite sets that we finite beings can observe.

So, your logic is based on trying to use an intuition about something 
that is impossible for you to experience, which just blows up your mind 
to smithereens by the contradiction you create in it.