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From: "Chris M. Thomasson" <chris.m.thomasson.1@gmail.com>
Newsgroups: sci.math
Subject: Re: Incompleteness of Cantor's enumeration of the rational numbers
 (extra-ordinary)
Date: Sat, 4 Jan 2025 12:38:54 -0800
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On 1/4/2025 12:48 AM, WM 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]
> 
> "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.

For me, there are infinitely many natural numbers, period... Do you 
totally disagree? Let's start small here... ;^)