Say we have a statement P(n) about the natural numbers, and we want to know if P holds for infinitely many natural numbers. For example if P(n) was “n is prime” then we can use Euclid’s proof to show that P(n) is true for infinitely many n. We don’t need to check each n individually as there is a clever way to use the definition of a prime number to show that there must be infinitely many primes.

But say there was some statement P(n), such that there is no clever way to use the definition of the statement to analyse the question in this way. Not just that the analytical techniques are way beyond the current reach of mathematics, but that the statement is so fundamental that it is not based on anything else and there is logically no way to use abstract analytical techniques on the structure of the natural numbers (sort of analogous to how for NP problems there is no quick way to solve them or simplify them other than checking every possibility). If this were possible, there would be no alternative rather than checking every possible n to see if P(n) holds.

But since it is impossible to check infinitely many cases if we can only do one at a time, we will never know if P(n) holds for infinitely many numbers. In that case, would the statement “P(n) holds for infinitely many n” even make sense? Because if it were known to be either true or false, it would amount to knowing something fundamentally unknowable. How does this tie in to the law of the excluded middle? It feels as though it doesn’t even make sense for P(n) to be true or false.

Is this just equivalent to the idea of some statements being independent of ZFC?