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

WM <wolfgang.mueckenheim@tha.de> wrote:

On 06.10.2024 12:16, Alan Mackenzie wrote:

 
You mean, that there is a difference?  I remain unconvinced.

[ .... ]

"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.

This idea of time may be what misleads the mathematically less adept
into believing that 0.999... < 1.

.... 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.  That is likely
why the terms have fallen out of use as mathematics has advanced.

Regards, WM

--
Alan Mackenzie (Nuremberg, Germany).