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Most natural numbers don't exist, for any useful definition of exist.

When you prove, say, by induction, that p(n) holds for any natural number n, and hear you teacher say that p(n) holds for all natural numbers, you start forming the idea that "all natural numbers" is a thing that exists. The set N, you think, surely by writing it, all natural numbers are called into existance.

And then, much later, you come upon problem, where actual existence of the number becomes better defined. Say, like finding a large prime. And suddenly "all numbers" becomes a confusing mental burden.