There are two mathematical concepts of infinity, potential infinity and actual infinity. I do not understand how the latter is being used. For the simplest infinite set, the natural numbers, we get:

Potential infinity: Every natural number that can be proven to be a prime number or not to be a prime number belongs to a finite initial segment that is followed by infinitely many natural numbers. An infinite set is much larger than every finite set. Therefore almost all natural numbers cannot be proven to be a prime number or not to be a prime number.

Actual infinity: Every natural number that can be proven to be a prime number or not to be a prime number belongs to a finite initial segment that is followed by infinitely many natural numbers. Nevertheless all natural numbers can be proven to be a prime number or not to be a prime number.

How is that possible?

Remark: This question had already been asked in SEMath and in MathOverflow. No answer was provided. The question was deleted.

closed as off-topic by Philip Klöcking Sep 14 at 18:49

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    It might be deleted here, too, but I think it is worth an answer. To show that a specific number is prime we would need to write it down and we can't write down all natural numbers. To show that all natural numbers are either prime or not does not require writing them all down and so an unambiguous answer can be provided. That might be the start of an answer. Welcome to this SE! – Frank Hubeny Sep 14 at 12:50
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    1. In contemporary mathematics, there is no formal distinction made between potential and actual infinity. These are outdated notions that no longer seem to be necessary. 2. Your proofs make no sense. – Dan Christensen Sep 14 at 12:57
  • @ Dan Christensen: I did not provide proofs. So you should perhaps first try to understand. – Uwe Sep 14 at 13:57
  • @Frank Hubeny: I am not interested in really proving primality. But when I asked the original question I used the phrase: "Every natural number that I can refer to" and was promptly advised that "refer to" is undefined. Therefore I use a phrase that every set theorist should be unable to not understand. – Uwe Sep 14 at 14:03
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    "Uwe" (user number 27663) is one of the twelve Mückenheim sock puppets (one of the two he named in honour of me), so it's not surprising that his "proofs" don't make sense. Mückenheim is bringing up the same nonsense over and over again. Experience shows that he is unwilling to learn. In fact, he's not really interested in answers, but rather in proselytizing, which means that he's constantly violating the rule from the help center: This site is not a personal blog or a pulpit for you to express your own personal philosophical beliefs. – Uwe Sep 14 at 14:12

Consider the following two questions about natural numbers to see how they differ.

  1. Is the natural number, n > 1, prime?
  2. Is the natural number, n > 1, prime or composite?

The first asks whether a specific natural number is prime. Current proofs for primality of an arbitrary natural number n > 1 require representing that number such that the representation gets arbitrarily large. Since numbers can require an arbitrarily large representation and our resources are finite, we can only answer that question for a finite number of natural numbers.

The second question asks something different. It wants to know if the only possibilities for a natural number, n > 1, are that it can be either prime or composite. It is not the case that there is a third option for these natural numbers. Also it is not the case that a natural number can be both prime and composite.

Answering this question does not require representing n in such a way that the representation gets arbitrarily large. It can be answered unambiguously for all natural numbers even though that set is arbitrarily large or "infinite".

Here is a proof of the result that natural numbers (or integers) greater than 1 have only two possibilities. They can be either prime or composite. There is no third option and they can't be both for the same number. This proof was provided by dotslash: https://math.stackexchange.com/q/441906/312852

Let n be any integer that is greater than 1. Consider all pairs of positive integers r and s such that n=rs. There exist at least two such pairs, namely r=n and s=1 and r=1 and s = n. Moreover, since n=rs, all such pairs satisfy the inequalities 1≤r≤n and 1≤s≤n. If n is prime, then the two displayed pairs are the only ways to write n as rs. Otherwise, there exists a pair of positive integers r and s such that n=rs and neither r nor s equals either 1 or n. Therefore, in this case 1

Note that this did not require using an arbitrarily large representation for any natural number n explicitly. The proof used variables: n, r and s.

Also note that the question is not trivial. If I did not restrict n so that it is larger than 1, but allowed n to be greater than or equal to 1, then there are three possibilities for an arbitrary natural number. It could be prime, composite or, if it happens to be 1, a unit.

The second question is different from the first. It is not trivial. And it can be answered unambiguously for the entire set of natural numbers.

With that preliminary consider the scenario presented by the OP:

Potential infinity: Every natural number that can be proven to be a prime number or not to be a prime number belongs to a finite initial segment that is followed by infinitely many natural numbers. An infinite set is much larger than every finite set. Therefore almost all natural numbers cannot be proven to be a prime number or not to be a prime number.

The number of elements that can be proven to be prime or not, that is, the number of natural numbers for which we can answer question 1, is finite given current algorithms. It is not "potentially infinite".

Actual infinity: Every natural number that can be proven to be a prime number or not to be a prime number belongs to a finite initial segment that is followed by infinitely many natural numbers. Nevertheless all natural numbers can be proven to be a prime number or not to be a prime number.

The last sentence in the quote is true. For all natural numbers greater than 1 the proof above showed that they are either prime or composite, that is, prime or not prime. One can check that when the natural number is 1 it is a unit and so not prime. The last sentence is true even though we can only prove, given current algorithms, that only a finite number of natural numbers are prime.

The question, How is that possible?, could be answered by saying that we are asking two different things about natural numbers. On the one hand, we want to know something specific about a natural number, its primality. On the other hand, we want to tell if the natural number can be only one of two types, prime or composite, and not something else and not both at the same time.

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    Your answer is completely fine and acceptable. Of course my main question is what you called no. 1. You say: "The number of elements that can be proven to be prime or not, that is, the number of natural numbers for which we can answer question 1, is finite given current algorithms." I agree with you, although I don't believe that there is an upper bound. That means potential infinity. Always infinitely many numbers will remain beyond the considered one. What I wish to learn is: How can we accept that (given infinite computing power) all numbers can be considered and checked? None remaining. – Uwe Sep 14 at 17:40
  • @Uwe As I see it that set will always be finite. That means there are even many finite sets of natural numbers whose members can never be all checked for primality. It is not just infinite sets that are problematic. However, if we have infinite computing power then anything goes. But that is not the world we live in. – Frank Hubeny Sep 14 at 18:50
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    "As I see it that set will always be finite." That is exactly my opinion too. The actually infinite set contradicts logic and in fact my question cannot be answered. It has been put "on hold" with the advice "you must have an actual, answerable question to go with them". "How is that possible that all natural numbers can be proven?" obviously has no answer. – Uwe Sep 14 at 19:55

You claim there are mathematical concept of "potential infinity" and "actual infinity", but there really aren't (at least not generally accepted ones). You're entitled to make up your own mathematical concepts (just don't complain if people don't adopt them themselves), but to say anything meaningful you need to define them. Your examples don't show any sort of clear distinction between the two. Math.SE and MathOverflow aren't going to pay attention to what they will see as a semi-random use of math-sounding words. You need to make some sort of mathematical statement, whether true or false, to be appropriate there.

It's obviously not possible to determine whether every single natural number is prime or composite (or 1, which is neither), since you can't list every number. It is mathematically possible to determine it for any given natural number, although it may be practically impossible. This is not just limited to a finite list. If you give me a finite list, I can name a number larger than any on your list, and it can be determined (in principle) whether it's prime or composite. "Infinite" means "unbounded", and you can't give me a largest natural number, so they're infinite. (

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    Of course there are generally accepted concepts of potential and actual infinity. See for instance Edward Nelson: "Hilbert's mistake" (2007) p. 3, Eric Schechter: "Potential versus completed infinity: Its history and controversy" (5 Dec 2009), D. Hilbert: "Über das Unendliche", Mathematische Annalen 95 (1925) p. 167 (translated in hs-augsburg.de/~mueckenh/Transfinity/Transfinity/pdf). The best statement has been given by G. Cantor: "In spite of significant difference between the notions of the potential and actual infinite, where the former is a variable finite (continued) – Uwe Sep 14 at 18:31
  • magnitude, growing above all limits, the latter a constant quantity fixed in itself but beyond all finite magnitudes, it happens deplorably often that the one is confused with the other." [Cantor, collected works, p. 374] Further the difference is clearly defined in my question for everybody who can understand mathematics. So please don't use your ingnorance or inability to understand as a measure. – Uwe Sep 14 at 18:32
  • You said: "you can't list every number". That is true only in potential infinity. In actual infinity there is a list containing every natural number. In technical terms it is called a bijection with |N and is the basic notion of transfinite set theory. – Uwe Sep 14 at 19:59

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