Re: Incompleteness of Cantor's enumeration of the rational numbers (extra-ordinary)

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

On 03.01.2025 22:38, Chris M. Thomasson wrote:

On 1/3/2025 9:09 AM, WM wrote:

On 03.01.2025 13:35, joes wrote:

Am Fri, 03 Jan 2025 09:39:01 +0100 schrieb WM:

Infinitely many can be removed without remainder. But only finitely
many
can be defined by FISONs.

It is very obvious there are infinitely many FISONs.

Obvious but only potentially infinite.

There are infinitely many FISONs. What in the heck do you mean by using
the word, "potentially"? It's as if you don't think infinity exists?

"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]

Philosopy.

"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..
[...] Cantor observed that many infinite sets of numbers are countable:
the set of all integers, the set of all rational numbers, and also the
set of all algebraic numbers. Then he gave his ingenious diagonal
argument that proves, by contradiction, that the set of all real numbers
is not countable. A consequence of this is that there exists a multitude
of transcendental numbers, even though the proof, by contradiction, does
not produce a single specific example." [T. Jech: "Set theory", Stanford
Encyclopedia of Philosophy (2002)]

Also philosophy.

"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]

E. Nelson is clearly not a mathematician.

According to (Gödel's) Platonism, objects of mathematics have the same
status of reality as physical objects. "Views to the effect that
Platonism is correct but only for certain relatively 'concrete'
mathematical 'objects'. Other mathematical 'objects' are man made, and
are not part of an external reality. Under such a view, what is to be
made of the part of mathematics that lies outside the scope of
Platonism? An obvious response is to reject it as utterly meaningless."
[H.M. Friedman: "Philosophical problems in logic" (2002) p. 9]

Possibly philosophy, more likely complete nonsense.

"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)]

Another philosopher?

"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)]

There may be a history to it, but there is no controversy, at least not
in mathematical circles.

The sequence of increasing circumferences (or diameters, or areas) of
circles is potentially infinite because the circumference of a circle
can become arbitrarily long, but it cannot be actually infinite because
then it would not belong to a circle. An infinite "circumference" would
have curvature zero, i.e., no curvature, and it could not be
distinguished what is the inner side and what is the outer side of the
circle.

The length of periods of decimal representations of rational numbers is
potentially infinite. The length is always finite although it has no
upper bound. The decimal representation is equal to a geometric series,
like 0.abcabcabc... = abc(10-3 + 10-6 + 10-9 + ...) which converges to
the limit  . A never repeating decimal sequence has an irrational limit..

More of the same

An interval of natural numbers without any prime number is called a
prime gap. The sequence of prime gaps assumes arbitrarily large
intervals but it cannot become actually infinite. None of the numbers n!
+ 2, n! + 3, n! + 4, ..., n! + n can be prime because n! = 123... n
contains 2, 3, ..., n as factors already. Therefore the set of gaps has
no upper bound. It is potentially infinite. It is not actually infinite
however, because there does not exist a gap with no closing prime number
because there is no last prime number.

The set of these prime gaps is infinite, without qualification.  Euclid
could have told you that.

Finally, the most familiar example is this: The (magnitudes of) natural
numbers are potentially infinite because, although there is no upper
bound, there is no infinite (magnitude of a) natural number.

There are no "actual" and "potential" infinity in mathematics.  The
notions are fully unneeded, and add nothing to any mathematical proof.
There is finite and infinite, and that's it.

When I did my maths degree, several decades ago, "potential infinity" and
"actual infinity" didn't get a look in.  They weren't mentioned a single
time.  Instead, precise definitions were given to "finite" and
"infinite", and we learnt how to use these definitions and what could be
done with them.

The only people who talk about "potential" and "actual" infinity are
non-mathematicians who lack understanding, and pioneer mathematicians
early on in the development of set theory who were still grasping after
precise notions.

Regards, WM

--
Alan Mackenzie (Nuremberg, Germany).