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

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]
 "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)]
 "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]
 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]
 "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."   [E. Schechter: "Potential versus completed infinity: Its history and controversy" (5 Dec 2009)]
 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.
 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.
 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.