When God, at the end of all time, will check what of his creation has been worthwile, he will also consider the set of natural numbers that ever have been used by his creatures. And he will find that only a very small subset has been applied. (This Idea goes back to Borel.)
For every usable number we have a finite set of predecessors and an infinite set of aleph_0 successors. So there is no usable natural number behind some borderline, although that border line cannot be determined yet.
Is it, in principle, possible to find circumstantial evidence for the existence of the aleph_0 inaccessible numbers - in order to satisfy platonists like Gödel? Or is postulating them by the axiom of infinity the only way to lay hold of them?