“In order to draw level with the tortoise, Achilles must “catch up” with it infinitely many times. Since this would require an infinite amount of time, he can never draw level with the tortoise.”It’s true that Achilles must catch up with the tortoise infinitely many times. But it’s false that this feat requires an infinite amount of time. Despite what is natural to think, Achilles can actually perform this infinite sequence of catch-ups in a finite amount of time.
“But the issue is not about time. The point is that, after any given catch-up, there’s always another one to be performed. So he can never perform them all.”Well, he can never perform them all if you mean that he cannot perform the last catch-up in the sequence, there being none. But he can certainly perform them all in a different sense. He can ensure that no catch-up, among the infinite lot, remains unperformed.
“But if no last catch-up is performed, it becomes incomprehensible how Achilles can ever transit out of the sequence of catch-ups. There he is, endlessly gunning for the tortoise’s last position, and then, suddenly, he is at the tortoise. How does he “pop out” of the endless sequence like that? There must be a last catch-up, at the end of which he draws level.”You must be thinking of Achilles performing the catch-ups in some idiosyncratic way. Thus, if you imagine him stepping in one dramatic bound to the tortoise’s last position each time, it will be impossible to grasp how he can draw level with the tortoise in any smooth way. So his eventual appearance beside the tortoise will seem like a miracle.