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

Moebius <invalid@example.invalid> wrote:

Am 05.01.2025 um 12:28 schrieb Alan Mackenzie:

"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.

Sure, though P. Lorenzen was an eminent mathematician who developed a
form of constructive mathematics (constructive analysis) and dialogical
logic.

See: https://en.wikipedia.org/wiki/Paul_Lorenzen
and: https://en.wikipedia.org/wiki/Dialogical_logic

Note that we can't be sure if Mückenheim's translation is accurate..

"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.

Sure. But T. Jech is a leading set theorist.

https://en.wikipedia.org/wiki/Thomas_Jech

"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.

Holy shit!

"Edward Nelson (May 4, 1932 – September 10, 2014) was an American
mathematician. He was professor in the Mathematics Department at
Princeton University. He was known for his work on mathematical physics
and mathematical logic. In mathematical logic, he was noted especially
for his internal set theory, and views on ultrafinitism and the
consistency of arithmetic. In philosophy of mathematics he advocated the
view of formalism rather than platonism or intuitionism."

https://en.wikipedia.org/wiki/Edward_Nelson

See: https://en.wikipedia.org/wiki/Internal_set_theory

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.

*sigh*

"Harvey Friedman (born 23 September 1948)[1] is an American mathematical
logician at Ohio State University in Columbus, Ohio. He has worked on
reverse mathematics, a project intended to derive the axioms of
mathematics from the theorems considered to be necessary. In recent
years, this has advanced to a study of Boolean relation theory, which
attempts to justify large cardinal axioms by demonstrating their
necessity for deriving certain propositions considered "concrete"."

https://en.wikipedia.org/wiki/Harvey_Friedman

"This chapter focuses on the work of mathematical logician Harvey
Friedman, who was recently awarded the National Science Foundation's
annual Waterman Prize, honoring the most outstanding American scientist
under thirty-five years of age in all fields of science and engineering..
Friedman's contributions span all branches of mathematical logic
(recursion theory, proof theory, model theory, set theory, and theory of
computation). He is a generalist in an age of specialization, yet his
theorems often require extraordinary technical virtuosity, of which only
a few selected highlights are discussed. Friedman's ideas have yielded
radically new kinds of independence results. The kinds of statements
that were proved to be independent before Friedman were mostly disguised
properties of formal systems (such as Gödel's theorem on unprovability
of consistency) or assertions about abstract sets (such as the continuum
hypothesis or Souslin's hypothesis). In contrast, Friedman's
independence results are about questions of a more concrete nature
involving, for example, Borel functions or the Hilbert cube."

Source:
https://www.sciencedirect.com/science/article/abs/pii/S0049237X09701545
(1985)

"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?

"Stephen George Simpson (born September 8, 1945) is an American
mathematician whose research concerns the foundations of mathematics,
including work in mathematical logic, recursion theory, and Ramsey
theory. He is known for his extensive development of the field of
reverse mathematics founded by Harvey Friedman, in which the goal is to
determine which axioms are needed to prove certain mathematical
theorems.[1] He has also argued for the benefits of finitistic
mathematical systems, such as primitive recursive arithmetic, which do
not include actual infinity."

Source: https://en.wikipedia.org/wiki/Steve_Simpson_(mathematician)

"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.

You are not familiar with "foundations of mathematics", right?

Moderately so.

Of course: "In practice, most mathematicians either do not work from
axiomatic systems, or if they do, do not doubt the consistency of ZFC,
generally their preferred axiomatic system."

Source: https://en.wikipedia.org/wiki/Foundations_of_mathematics

There are no "actual" and "potential" infinity in mathematics.

If you say so. :-P

None of the above extracts is about mathematics.  If there were such
things as "potential" and "actual" infinity in maths, then they would
make a difference to some mathematical result.  There would be some
theorem provable given the existence of PI and AI which would be false or
unprovable with just plain infinite, or vice versa.  Or something like
that.  Nobody in this discussion has so far attempted to cite such a
result.

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.

Sure. But this needs a context; usually (some sort of) set theory.

The only people who talk about "potential" and "actual" infinity are
non-mathematicians who lack understanding

Like those mentioned above?

OK, and mathematicians in their time off.  ;-)

--
Alan Mackenzie (Nuremberg, Germany).