Goldbach Conjecture and Busy Beaver numbers?
The statement is about infinitely many numbers, but its demonstration (or refutation) would have to be a finite exercise. If possible.

The surprise may come from the (false) assumption that finding BB(100) would be a "theoretically easier" problem, only made impossible for practical reasons - since there are so many machines, and they can run for such a long time before stopping, if at all - after all, they're just machines...

The truth is that discovering BB(n), for n large enough, has to be an unsurmountable task, for both theoretical and practical reasons.