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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 17:26:07 +0200
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On 06.10.2024 16:52, Alan Mackenzie wrote:
> WM <wolfgang.mueckenheim@tha.de> wrote:

>> "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, ....
> 
> That is a mistake.  The ellipses indicate the enumeration.  There is no
> time.  If one must consider time, then the enumeration happens
> instantaneously.

In potential infinity there is time or at least a sequence of steps.
> 
> This idea of time may be what misleads the mathematically less adept
> into believing that 0.999... < 1.

That is true even in actual infinity.
We can add 9 to 0.999...999 to obtain 9.999...999. But multiplying 
0.999...999 by 10 or, what is the same, shifting the digits 9 by one 
step to the left-hand side, does not increase their number but leaves it 
constant: 9.99...9990.

10*0.999...999 = 9.99...9990 = 9 + 0.99...9990 < 9 + 0.999...999
==>  9*0.999...999 < 9 as it should be.
> 
>> .... 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)]
> 
> The above is all very poetic, this supposed difference between "actual"
> and "potential" infinite, but it is not mathematical.  There are no
> mathematical theorems which depend for their theoremhood on the supposed
> distinction between "actual" and "potential" infinite.

Set theory depends on actual infinity. Bijections must be complete. But 
Cantor's bijections never are complete. Cantor's list must be completely 
enumerated by natural numbers. The diagonal number must be complete such 
that no digit is missing in order to be distinct from every listed real 
number. Impossible. All that is nonsense.

Regards, WM